Trigonometry made easy

Summary

This Tech Math Channel video introduces trigonometry, focusing on its application to right-angle triangles. The host explains the fundamental relationships between triangle sides and angles, using the example of an angle labeled Theta. Key concepts include labeling sides as opposite, adjacent, and hypotenuse, and using trigonometric functions like sine, cosine, and tangent to solve for unknown angles or side lengths. The video simplifies complex concepts with mnemonics and practical examples, demonstrating how to calculate and use inverse functions on a calculator to find angles from side lengths.

Takeaways

• 📚 Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of a right-angle triangle.
• 🔢 The three primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan), which relate the angles and side lengths of a right-angle triangle.
• 📐 The sides of a right-angle triangle are labeled as the hypotenuse (opposite the right angle), the opposite side (opposite the angle in question), and the adjacent side (next to the angle in question).
• 💡 The sine function is defined as the ratio of the opposite side to the hypotenuse, cosine as the adjacent side to the hypotenuse, and tangent as the opposite side to the adjacent side.
• 🧠 Memorizing the acronym SOHCAHTOA can help recall the trigonometric functions: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
• 🔄 To solve for an unknown side length, one can use the appropriate trigonometric function based on the given angle and known side lengths.
• 🔢 Using a calculator, one can find the value of the trigonometric functions for specific angles and then solve for the unknown side lengths.
• 🔄 To find an unknown angle given two side lengths, use the inverse trigonometric functions (e.g., sin^-1, cos^-1) available on most calculators.
• 📝 The process involves labeling the sides, selecting the correct trigonometric function, substituting the known values, and then solving for the unknown using basic algebraic manipulations.
• 👨‍🏫 The video script provides practical examples demonstrating how to apply trigonometry to solve for unknown sides and angles in right-angle triangles.

Q & A

What is the main focus of the video?

The main focus of the video is to explain trigonometry, specifically its application to right-angle triangles, and how it can be used to find unknown angles or side lengths.

What are the three sides of a right-angle triangle called?

The three sides of a right-angle triangle are called the hypotenuse, the opposite side, and the adjacent side.

What are the three primary trigonometric functions?

The three primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan).

How are the trigonometric functions related to the sides of a right-angle triangle?

The trigonometric functions relate to the sides of a right-angle triangle as follows: sine is the ratio of the opposite side to the hypotenuse, cosine is the ratio of the adjacent side to the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side.

What is the mnemonic provided in the video to remember the trigonometric functions?

The mnemonic provided in the video to remember the trigonometric functions is 'SOHCAHTOA', which stands for Sine equals Opposite over Hypotenuse, Cosine equals Adjacent over Hypotenuse, and Tangent equals Opposite over Adjacent.

How does the video suggest solving for an unknown side length using trigonometry?

The video suggests first labeling the sides as opposite, adjacent, or hypotenuse, then choosing the appropriate trigonometric function based on the given angle and known side, and finally substituting the values into the function to solve for the unknown side length.

What is the process for finding an unknown angle when two side lengths are known?

To find an unknown angle when two side lengths are known, one must first label the sides, then use the appropriate trigonometric function (sine or cosine), calculate the ratio, and finally use the inverse function on a calculator to find the angle.

How does the video demonstrate solving for an unknown side length using the sine function?

The video demonstrates solving for an unknown side length using the sine function by providing an example with a 35° angle and a hypotenuse of 12 units, then using the sine function to calculate the opposite side length as 6.88 units.

What is the role of the inverse trigonometric functions in solving for angles?

The inverse trigonometric functions, such as sin^-1 or cos^-1, are used to find the angle when the ratio of the sides is known. They are essential for converting back from the ratio to the actual angle measure.

How does the video recommend practicing and mastering trigonometry?

The video recommends practicing and mastering trigonometry by solving various problems, understanding the relationships between the sides and angles, and getting familiar with the functions and how to use them on a calculator.

Outlines

00:00

📚 Introduction to Trigonometry

The video begins with a warm welcome to the Tech Math channel, where the presenter introduces the topic of trigonometry. Trigonometry is described as a branch of mathematics that focuses on the relationships between the sides and angles of a right-angle triangle. The presenter explains that trigonometry allows us to use one angle and a side length to determine other side lengths or unknown angles. The concept of labeling the sides of a triangle as 'opposite', 'adjacent', and 'hypotenuse' relative to an angle (Theta) is introduced. The video then explains the three main trigonometric functions: sine (opposite/hypotenuse), cosine (adjacent/hypotenuse), and tangent (opposite/adjacent), using the mnemonic 'SOHCAHTOA' to help remember these relationships. The presenter emphasizes the practicality of trigonometry in solving problems involving unknown side lengths and angles.

05:02

🔍 Solving Trigonometric Problems

This paragraph delves into solving trigonometric problems using the labeled sides of a right-angle triangle. The presenter demonstrates how to find unknown side lengths by applying the appropriate trigonometric function based on the given angle and known side. The process involves substituting the known values into the trigonometric function and solving for the unknown. The video provides two examples: one where the sine function is used to find the length of the opposite side given the hypotenuse and angle, and another where the tangent function is used to find the adjacent side length. The presenter also shares a helpful trick for solving these equations by creating a 'friendly equation' to assist in isolating the variable. The examples are worked through step by step, showing how to calculate and interpret the results, with an emphasis on the practical application of trigonometry.

10:03

🧮 Finding Angles from Side Lengths

The final paragraph of the script focuses on the process of finding angles when the side lengths of a right-angle triangle are known. The presenter explains that once the sides are labeled as opposite, adjacent, or hypotenuse, the appropriate trigonometric function (sine, cosine, or tangent) can be used to find the unknown angle. The video provides an example where the sine function is used to determine the angle given the lengths of the opposite and hypotenuse sides. The presenter also demonstrates how to use the inverse trigonometric functions on a calculator to find the angle, highlighting the importance of knowing how to navigate calculator functions for these calculations. The video concludes with an invitation for viewers to engage with the content, suggesting that more videos on trigonometry will be made if there is interest, and encourages viewers to like and subscribe for more educational content.

Keywords

💡Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles, particularly right-angled triangles. In the video, trigonometry is the central theme, as it is used to explain how to calculate unknown angles and side lengths using the properties of right-angled triangles. The script mentions that trigonometry allows us to use one known angle and side length to determine other unknown elements within the triangle.

💡Right Angle Triangle

A right angle triangle is a triangle with one angle measuring 90 degrees. This type of triangle is fundamental to trigonometry because the relationships between its sides and angles are predictable and can be expressed through trigonometric functions. The video uses right angle triangles to demonstrate how to apply trigonometric ratios and functions to solve for unknowns.

💡Hypotenuse

The hypotenuse is the longest side of a right angle triangle, opposite the right angle. In the video, the hypotenuse is one of the sides labeled in the triangle diagrams, and it is used in the calculations involving trigonometric functions, such as sine and cosine, where it serves as the reference for the ratios.

💡Opposite

In the context of trigonometry, the opposite side is the side of a right angle triangle that is opposite the angle being considered. The video explains that the sine function, for example, is the ratio of the opposite side to the hypotenuse, which is crucial for solving problems involving unknown side lengths.

💡Adjacent

The adjacent side is the side of a right angle triangle that is next to (or adjacent to) the angle being considered. The video mentions that the cosine function is the ratio of the adjacent side to the hypotenuse, which is used to find unknown angles when the side lengths are known.

💡Sine Function

The sine function, often abbreviated as 'sin', is one of the primary trigonometric functions. It is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right angle triangle. The video uses the sine function to demonstrate how to calculate unknown side lengths when the angle and the hypotenuse are known.

💡Cosine Function

The cosine function, abbreviated as 'cos', is another fundamental trigonometric function. It is the ratio of the length of the adjacent side to the length of the hypotenuse in a right angle triangle. The video script uses the cosine function to show how to find an unknown angle when the lengths of the adjacent and hypotenuse sides are known.

💡Tangent Function

The tangent function, denoted as 'tan', is the ratio of the length of the opposite side to the length of the adjacent side in a right angle triangle. The video explains the tangent function as part of the set of trigonometric functions used to solve for unknown sides or angles in a triangle.

💡SOHCAHTOA

SOHCAHTOA is a mnemonic used to remember the basic trigonometric functions: Sine, Opposite over Hypotenuse; Cosine, Adjacent over Hypotenuse; Tangent, Opposite over Adjacent. The video script introduces this mnemonic as a tool to help viewers remember the definitions of the sine, cosine, and tangent functions.

💡Inverse Trigonometric Functions

Inverse trigonometric functions, such as arcsin, arccos, and arctan, are used to find the angle when the ratio of the sides of a right angle triangle is known. The video script demonstrates the use of inverse functions on a calculator to determine angles from known side lengths, which is crucial for solving certain trigonometric problems.

Highlights

Introduction to trigonometry and its focus on the relationships in a right-angled triangle.

Explanation of how trigonometry uses side lengths and angles to solve for unknowns.

Labeling the sides of a right-angled triangle as opposite, adjacent, and hypotenuse.

Definition of the sine function as the ratio of opposite to hypotenuse.

Definition of the cosine function as the ratio of adjacent to hypotenuse.

Definition of the tangent function as the ratio of opposite to adjacent.

Mnemonic 'SOHCAHTOA' to remember the trigonometric functions.

Step-by-step guide to solving for an unknown side using sine in a right-angled triangle.

Using a calculator to find the sine of an angle and solving for an unknown side length.

Example problem solving where an angle and one side length are known, and another side length is unknown.

Guide on how to use the tangent function to find an unknown side length.

Demonstration of solving for an angle when two side lengths are known.

Using the inverse sine function on a calculator to find an angle from a ratio.

Another example of finding an angle using the cosine function and an inverse function on a calculator.

Emphasis on the importance of knowing how to use a calculator for trigonometric functions.

Encouragement for viewers to ask questions and engage with the content for further assistance.

Invitation to like, subscribe, and support the TechMath channel for more educational content.

Transcripts

gday welcome to the tech math Channel

what we're going to be having a look at

in this video is trigonometry so sit

back and learn all about it and if you

like the video please remember hit the

like button beneath the video there and

subscribe to the techmath channel so

trigonometry deals with this particular

shape here a right angle triangle and

what it does is it's a branch of

mathematics that studies the

relationships between the sides of this

triangle and the angles that occur

within it okay so pretty much we can use

say angle here and a side length to work

out other side lengths or we could use t

two side lengths here to work out

unknown angles that's what trigonometry

allows us to do so how does it do this

well it's fairly simple if we were to

consider say an angle here in this

triangle so I'm just going to put this

down and this angle here is called Theta

pretty much what it's saying is this for

this particular angle here in a right

angle triangle in this particular

location in that right angle triangle

these two side lengths here would have a

particular ratio they would always be an

equivalent length compared to one

another okay this length and this length

would have a certain ratio and this

length and this length would have a

certain ratio and trigonometry uses this

to be able to work out unknown side

lengths and unknown angles so how do we

do this well the first thing we have to

do is we have to be able to label the

sides of this particular triangle so in

this particular triangle you're going to

notice we've got a right angle here we

have this angle Theta which we've uh

we've already labeled here we also have

three sides here we have this longest

side here the longest side is called the

hypotenuse I'm going to write that in

the high pot and use I'm going to put

that down as a

h here we have the opposite side I'll

write that over here the

opposite what do I mean by that this

particular side is opposite feeter we

put that down as an

O along this particular side this

remaining side which is next to feta we

have the adjacent adjacent okay adjacent

means next to and we label that with an

A so now we've done that as I was saying

all these side lengths here the opposite

the adjacent the hypotenuse all have

particular ratios to one another based

on whatever this particular angle here

is okay so there's three different

functions we are thinking about when

we're thinking about these ratios

because we have three different ways we

could compare the sides we could be comp

comparing these two sides to feta or

these two sides or you can be comparing

these two sides and our three main

trigonomic functions are as follows we

have the sign function which is the

ratio of the opposite and the hypotenuse

we have the coine function which is the

ratio between the adjacent and the

hypotenuse and we have the tangent

function which is the ratio between the

opposite and the adjacent function

function now there's a really really

easy way we can remember these uh when

we're doing these and this is as follows

I'll write this pum monic down right now

and here it is some old hags can't

always hack their old age okay

so sign equals opposite over hypotenuse

can't always hack cos equals adjacent

over hypotenuse their old age tan equals

opposite over adjacent so when I was

solving a trigonomic equation pretty

much the very first thing I'd do is what

we did first off here I'd label these

unknown sides the next thing I'd do is I

determine which trigonomic function I

was going to use so we're pretty much

all set now to solve some trigonomic uh

problems so let's do that so for our

first example here we have a right angle

triangle okay it has an angle of 35° it

has one side length of 12 M and another

unknown side length which we're going to

be trying to work out so the very first

step to work out this unknown side

length is is we are going to do like we

do with any trigonomic equation or any

trigonomic problem we are going to label

the unknown sides so first off we have

this long side here which is the

hypotenuse then we have this side which

is opposite this angle opposite this 35

here this is the opposite so which of

our trigonomic functions deals with the

opposite and the hypotenuse and you're

going to see that it's sign here s is

equal to opposite over hypotenuse some

old hags so I'm going to write this down

s

Theta is equal to the opposite over the

hypotenuse and now what we do is we just

go through and substitu in our values so

sin Theta this is sin

35° is equal to the opposite the

opposite is what we're trying to work

out here x so I'll put that in as X over

the hypotenuse which is

12 so we can now work this out a little

bit further we could actually say okay

uh sin 35 we put that into a calculator

we're going to get the answer is

0.57 which is equal to x/

12 what can we do now so what we have to

do is we have to get X by itself okay so

X is going to be equal to what now

there's a little trick I use here this

may or may not help you you may or may

not like it okay I'm sure I'm going to

get plenty of hate for this but what I

do is this when I'm not certain what to

do here and I'm trying to solve this

particular problem here I just write up

an equation next to it a friendly

equation as it were the equation I'm

going to write is this one 3 = 6 over 2

and we're trying to deal with this

particular value here the value up here

so what would you do with three and two

to get six well you'd multiply them so

we're going to multiply these two

numbers 12 *

0.57 so 12 * 0.57 and we'll get our

answer so if you do that what answer do

you get you get our answer of

6.88 M okay so this side length this

opposite is 6.88 M and that's how easy

trigonometry is to use okay so we're

going to go through another example and

then I'm going to go through an example

where I look at how to work out the

angle from uh 29 side length so it's

it's a bit of a tweak here so stay tuned

for that one as well okay but let's just

go through another one of these type

examples okay for our second example

let's have a look we have a right angle

triangle we have an angle of 48° we know

that this side length here is 15 and

we're trying to work out this unknown

side length here so let's label our

sides first we have this particular side

here which is opposite the angle here so

that's the opposite we know that this

one here is the hypotenuse that's the

easy one to spot so it leaves this one

here being the adjacent okay and it

makes sense it's the shorter one that's

running next to the angle here so which

one of these functions uses opposite and

adjacent you're going to see here is tan

tan Theta equal opposite over adjacent

so let's sub in our values now so tan

feta becomes tan

48° which is equal to the opposite which

is 50 m over our unknown our

X okay we can put tan 48 into the

calculator if you do this you're going

to get this answer of 1 .11 okay the

opposite adjacent have that particular

ratio of 1.11 for an angle of 48° which

is equal to 15 / X so now to solve for x

and if you're not certain what to do you

might know this straight away but you

could do this once again you could go

okay 3 = 6 / 2 and we're trying to work

out the value on the bottom here the two

so that would be 6 / by 3 this number

divid by this number this number divided

by this number x x here here is going to

be equal to this number ID by this

number 15 /

1.11 which is equal to how much

1.51

M okay so that's how that particular

type of our function in trigonometry

works it's pretty simple right now we're

going to go through some examples we're

going to look at how to work out the

angle from s and side lengths it's

fairly simple there's just a couple of

tweaks with this so in this example here

we have a right angle triangle and we

know two side lengths we know that this

side length here is 105 M and we know

that this side length here is 33 M what

we're trying to find out is we're trying

to find out this unknown angle that

would accommodate these side lengths so

how do we do that well it's just one

little variant and I'll get to that as

we uh do this particular problem the

very start though is exactly the same we

are just going to go through and label

whether our sides are opposite

hypotenuse or adjacent so we know this

long s side here is going to be

the hypotenuse we know that this side

opposite angle Theta here is the

opposite so which one of the functions

are we dealing with this is our second

thing we can deal with which function

and we're going to be dealing with sign

here

s Theta is going to equal the opposite

over the hypotenuse so what is the

opposite over the hypotenuse we're going

to see here that we have the opposite

which is 33 me

over 105

M okay so sin Theta is equal to 33 over

105 if we' have work this out what's 33

/ 105 you're going to see that sin Theta

is equal to

0.314 okay that's just the matter of

going 33 / 105 and we get this answer

here so what we do now is just a little

variant because we have to actually go

back we got the ratio we're trying to go

back to the angle and you're going to

notice on calculators that there's

either a second function or something

like that that allows you to go from s

to this particular thing

s-1 okay we want to be using that here

we're going to be hitting 3 0.314 and

we're going to hit s-1 or second

function sign here if we do that we're

going to get the answer of theta being

equal to 18.3

okay so just make sure you know how to

do that on your calculator okay uh

anyway we'll go through one more of

these

examples okay for this example here we

have a right angle triangle we have two

side LS we know we know that this one

here is 17 we know this one here is 12

and we're trying to work out the angle

that accommodates these so let's go

through and do this uh the very first

thing we do is we're going to label our

sides we have the hypotenuse which you

can see we're not going to be deal

dealing with the opposite we're in fact

dealing with the adjacent so which one

of these functions are we dealing with

and you're going to see here the

adjacent the hypotenuse is the cosine

function so cos Theta is equal to the

adjacent over the

hypotenuse okay so what is that going to

be cos Theta which is what we're trying

to find out is equal to the adjacent

which is

12 over the hypot years which is 17 so

we work out what 12 / 17 is we get the

answer of

0.71 okay cos Theta is equal to

0.71 so we're going to be not working

out cos we're going to be working out

the inverse of cos cos to the ne1 so

you're going to hit second function cos

and you're going to get uh when you do

that you're going to get the answer for

Theta Theta or our angle here is equal

to

45.1 de

so that's how you go doing trigonometry

and it's most basic it's pretty simple

right it's just those tweaks there and

it's also getting to know the calculator

that you are using so anyway hopefully

that video is some help to you if you

got any problems please let me know and

I'll make some more videos on

trigonometry I'm sure I'm going to have

some issues where people are going to

get stuck with these please if you like

the video remember like And subscribe

hey and below the video in the

description there there is the uh

patreon feed there you can always uh

subscribe but you can also actually

donate to the techmath channel on a

video by video basis so we can keep

plugging these videos and making more

and more and more and more of them

anyway thanks for watching we'll see you

next time

bye