Finding Angles - Trigonometry in Right-angled Triangles - Tutorial / Revision (4/5)

Summary

This tutorial introduces the use of inverse trigonometric functions to find missing angles in right-angled triangles. It guides viewers through labeling sides, identifying active sides, and applying formulas like tan and cos with their inverses to solve for angles. The instructor demonstrates using a calculator to find angles, rounding the results to 41.0 and 72.2 degrees, respectively. The video concludes with an invitation for questions and a comparison of Pythagoras and trigonometry for future lessons.

Takeaways

• 📚 This tutorial is part of a series on trigonometry, focusing on using inverse trigonometric functions to find missing angles in right-angled triangles.
• 🔍 The presenter suggests reviewing previous tutorials if one is not familiar with trigonometric formulas, indicating the importance of foundational knowledge.
• 🧭 The tutorial introduces inverse trigonometric functions: sin⁻¹, cos⁻¹, tan⁻¹, and how they are accessed on calculators, which is crucial for solving the problems.
• 📐 The first step in solving the problem is labeling the sides of the triangle, which helps in identifying the 'active' sides needed for the calculations.
• 🔢 The tutorial demonstrates using the tangent function (tan) to find an angle when given the lengths of the opposite and adjacent sides.
• ✅ The use of the inverse tangent function (tan⁻¹) is shown to isolate and find the value of the angle, with an example calculation provided.
• 📉 The process of substituting values into the trigonometric formula and solving for the angle is explained step by step, emphasizing precision.
• 📈 Another example is given using the cosine function (cos) to find an angle when given the lengths of the adjacent side and the hypotenuse.
• 🔄 The inverse cosine function (cos⁻¹) is used similarly to tan⁻¹ to find the angle, with an example calculation shown.
• 📝 The importance of writing out the formulas in full is highlighted to ensure correct substitution and avoid errors.
• 🤔 The tutorial ends with an invitation for questions and a reminder of the availability of further resources for understanding trigonometry and its applications.

Q & A

What is the main topic of this tutorial?

The main topic of this tutorial is using trigonometry to find missing angles in right-angled triangles.

What is the series number of this tutorial in the trigonometry series?

This is tutorial number 4 in the trigonometry series.

What are the inverse trigonometric functions mentioned in the script?

The inverse trigonometric functions mentioned are sin^(-1), cos^(-1), and tan^(-1).

How can you access the inverse trigonometric functions on a calculator?

On a calculator, these functions are usually accessed by pressing shift or sometimes 2nd F and then the sin, cos, or tan buttons.

What are the active sides in a triangle when using trigonometric ratios?

The active sides are the two sides given in the problem statement, which are used in the trigonometric formula to find the missing angle.

What trigonometric ratio uses the opposite and adjacent sides of a right-angled triangle?

The tangent ratio (tan) uses the opposite and adjacent sides of a right-angled triangle.

How do you find the angle X using the tangent ratio?

To find the angle X, you use the inverse tangent function, tan^(-1), on the calculator with the opposite side divided by the adjacent side.

What is the result of the first example calculation in the script?

The result of the first example calculation is 41.0 degrees for the angle X.

What trigonometric ratio uses the adjacent side and the hypotenuse of a right-angled triangle?

The cosine ratio (cos) uses the adjacent side and the hypotenuse of a right-angled triangle.

How do you find the angle Y using the cosine ratio?

To find the angle Y, you use the inverse cosine function, cos^(-1), on the calculator with the adjacent side divided by the hypotenuse.

What is the result of the second example calculation in the script?

The result of the second example calculation is 72.2 degrees for the angle Y.

What is the purpose of the tutorial mentioned at the end of the script?

The purpose of the mentioned tutorial is to compare Pythagoras' theorem with trigonometry and to determine when to use each.

Outlines

00:00

📚 Introduction to Trigonometry Tutorial

This paragraph introduces the trigonometry tutorial focused on finding missing angles in right-angled triangles using inverse trigonometric functions. It is the fourth tutorial in a series and assumes familiarity with basic trigonometric formulas. The speaker encourages viewers to catch up on previous lessons and mentions the use of calculator functions like sin^-1, cos^-1, and tan^-1 for solving these problems. The paragraph sets the stage for a step-by-step approach to solving trigonometric problems.

🔍 Understanding and Labeling Sides in Triangles

This paragraph delves into the process of identifying and labeling the sides of a right-angled triangle, which are essential for applying trigonometric formulas. The speaker explains the importance of recognizing 'active sides' and provides a step-by-step guide on how to proceed with the problem-solving process. It emphasizes the need to label sides correctly before selecting the appropriate trigonometric formula to find the missing angle.

📐 Applying Trigonometric Formulas to Find Angles

The speaker illustrates the application of trigonometric formulas to find missing angles, specifically using the tangent function in this example. The paragraph explains how to set up the formula with given side lengths, substitute the values, and use the inverse tangent function to solve for the angle. It provides a clear demonstration of how to calculate the angle using a calculator and emphasizes the importance of entering values correctly to avoid errors.

🧭 Using Cosine to Solve for Another Angle

This paragraph continues the tutorial by showing how to use the cosine function to find another angle in a right-angled triangle. The process involves identifying the adjacent and hypotenuse sides, writing out the cosine formula, and then using the inverse cosine function to solve for the angle. The speaker provides a practical example, demonstrating the calculation and rounding off the result to find the angle's measure.

📘 Conclusion and Further Resources on Trigonometry

The final paragraph wraps up the tutorial by summarizing the essential aspects of right-angle trigonometry covered in the video. It mentions an upcoming tutorial comparing Pythagorean theorem with trigonometry and when to use each. The speaker invites viewers to ask questions or comment on the YouTube channel or website, promising to assist with any difficulties, and concludes the tutorial with thanks and farewell.

Keywords

💡Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles, particularly right-angled triangles. In the context of this video, trigonometry is used to find missing angles in right-angled triangles using the ratios of the sides. The script mentions trigonometry as the main theme of the tutorial series, emphasizing its importance in solving various geometric problems.

💡Right-Angled Triangles

A right-angled triangle is a triangle that has one angle measuring exactly 90 degrees. The script focuses on using trigonometry to find missing angles in such triangles. The concept is central to the video as it provides the foundation for understanding how to apply trigonometric functions to solve for unknown angles.

💡Inverse Trigonometric Functions

Inverse trigonometric functions, such as sin^-1, cos^-1, and tan^-1, are used to find the angle when the ratio of the sides of a right-angled triangle is known. The script explains that these functions are accessed on a calculator by pressing specific keys, and they are essential for solving the problems presented in the tutorial.

💡Sine (sin)

Sine is a trigonometric function that relates the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. In the script, the inverse sine function, sin^-1, is used to find the angle when the opposite and hypotenuse sides are known, as demonstrated in the calculation example.

💡Cosine (cos)

Cosine is another trigonometric function that represents the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle. The script mentions the inverse cosine function, cos^-1, to find an angle when the adjacent side and the hypotenuse are given, as shown in the second example.

💡Tangent (tan)

Tangent is the ratio of the length of the opposite side to the adjacent side in a right-angled triangle. The script refers to the tangent function when discussing the active sides in the triangle and how to use the inverse tangent function, tan^-1, to find an angle.

💡Calculator

A calculator is an electronic device used for performing mathematical calculations. The script provides instructions on how to use a calculator to find inverse trigonometric functions, emphasizing the importance of knowing where to find these functions on different types of calculators.

💡Active Sides

In the context of the script, 'active sides' refers to the sides of the triangle that are involved in the trigonometric calculations to find the missing angle. The script explains how to identify these sides and use them in the trigonometric formulas.

💡Formula

A formula in mathematics is a concise way of expressing information symbolically as a relationship between quantities. The script uses trigonometric formulas to demonstrate how to set up and solve for the missing angles in right-angled triangles.

💡Hypotenuse

The hypotenuse is the longest side of a right-angled triangle, opposite the right angle. The script mentions the hypotenuse in the context of using cosine and sine functions to find angles when the hypotenuse and one other side are known.

💡Pythagoras

Pythagoras refers to the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. The script mentions a comparison between Pythagoras and trigonometry in a separate tutorial, indicating the broader context of geometric problem-solving.

Highlights

Introduction to the tutorial on using trigonometry to find missing angles in right-angled triangles.

This is the fourth tutorial in the series on trigonometry.

Encouragement to review previous tutorials for a refresher on trigonometric formulas.

Explanation of the use of inverse trigonometric functions like sin^-1, cos^-1, and tan^-1.

Instructions on how to access inverse trig functions on calculators.

The importance of knowing where to find the inverse trig functions on various devices.

Step-by-step guide on labeling sides in a triangle for trigonometry problems.

Identification of active sides in a triangle for solving trigonometry problems.

Using the tangent function (tan) when given two side lengths to find an angle.

Substitution of values into the tangent formula to solve for the angle.

Application of the inverse tangent function (tan^-1) to find the angle.

Demonstration of entering values on a calculator to find the angle using tan^-1.

Example calculation resulting in an angle of 41.0 degrees.

Introduction of another example using the cosine function (cos).

Explanation of using the inverse cosine function (cos^-1) to find an angle.

Example calculation using cos^-1 resulting in an angle of 72.2 degrees.

Conclusion of the essential parts of right-angle trigonometry covered in the tutorial.

Invitation for questions and comments on YouTube or the website for further assistance.

Closing remarks and thanks for watching the tutorial.

Transcripts

hi again and welcome to this me versus

math tutorial which is about using

trigonometry to find missing angles in a

right angled triangles this is tutorial

4 in our series on trig so if you've got

some catching up to do first then please

do that otherwise let's get this show on

the road hopefully the trig formulas are

starting to look really familiar to you

by now in fact you're probably getting a

bit sick of them but hey you've probably

started to memorize them so it's all

good just before we get going properly

for this tutorial we're going to be

using what are known as the inverse trig

functions

these look like sin minus 1 cos minus 1

tan minus 1 and so on on a calculator

these are normally accessed by pressing

shift or sometimes 2nd F and then the

sin cos or tan buttons it's often

similar if you're using a calculator on

your phone tablet or other device - or

sometimes they can actually have their

own button on those sorts of devices so

check that you know where to find those

3 functions this question then is asking

us to find a missing angle apart from

that fact it looks fairly similar to the

sorts of questions we were looking at in

a previous tutorial and most of our

actions are actually going to be exactly

the same as they were when we find an

aside so step one as previously label

your sides first now the next step is to

identify what I've been referring to is

the active sides in our triangle

previously one of these was a side which

we were given but one was the side that

we were looking for this time we have

been given two side lengths so these are

our two active sites let's take them so

we're looking for the formula that uses

the opposite and adjacent which is Toa

still just as before again let's now

write our formula out in full

well let's substitute our values in two

we don't know the angle but it is called

X in our question so

gonna replace the theta with an X now

always be really careful that you get

the next two numbers the right way

around the opposite should be on top so

it's 8.6

divided by the adjacent 9.9 now we have

tan x and we just want X so what we

actually have to do now is to cancel out

10 to do this we have to use the inverse

of tan or tan -1 which I spoke about

previously finding on your calculator so

X is actually equal to inverse tan of

8.6 divided by 9.9 again this is the

value of a good dedicated calculator if

you've got one you can enter everything

on the right-hand side now in one go you

use the fraction button if you wish and

just as before you'll get an answer we

need to round it off ultimately we get

41 point 0 degrees let's look at just

one more example as before let's label

our sides identify the two active sides

and as we know 4 angles this is the two

sides we've been given we're going to be

using cars as that uses the adjacent and

the hypotenuse always write the formula

out in full it just helps make sure

everything is substituted into the right

place so cos theta is equal to adjacent

over hypotenuse so cos y is equal to

three point eight over twelve point four

the adjacent on the top the hypotenuse

on the bottom to get Y on its own we

have to get rid of that cause so we need

to use the inverse of cars or cos minus

1 so Y is going to be equal to inverse

cos of three point eight over twelve

point four that's before get a good

calculator where you can enter all this

in one go get an answer and round it up

to give 72 point two degrees

okay that's angles done and that is all

the essential parts really of right

angle trigonometry we've got another

tutorial where we compare Pythagoras in

trigonometry and look at which to use

and when any questions on anything we've

done then as always just supposed to

comment either on YouTube or at the

website at me versus mass comm and I'll

do my best to help as always thanks for

watching take care and bye for now

you