Sin Cos Tan

Summary

This video tutorial simplifies the understanding of the six basic trigonometric functions using the mnemonic 'sohcahtoa'. It breaks down the sine, cosine, and tangent by relating them to the sides of a right-angled triangle, labeling the hypotenuse, opposite, and adjacent sides accordingly. The script then calculates these functions for a given angle 'a', and demonstrates how to derive the remaining three functions—cosecant, secant, and cotangent—by inverting the ratios of the initial three, providing a clear and memorable method for students to grasp these mathematical concepts.

Takeaways

• 📚 The video teaches the six basic trigonometric functions using the mnemonic 'sohcahtoa'.
• 📝 'Soh' stands for sine, which is the ratio of the opposite side to the hypotenuse.
• 📝 'Cah' stands for cosine, representing the ratio of the adjacent side to the hypotenuse.
• 📝 'Toa' stands for tangent, which is the ratio of the opposite side to the adjacent side.
• 📐 The video demonstrates how to label the sides of a right triangle for trigonometric calculations.
• 📏 The hypotenuse is the longest side and is found opposite the 90-degree angle.
• 📏 The opposite side is the one directly opposite the angle in question.
• 📏 The adjacent side is the one touching the angle but not the hypotenuse.
• 🔢 The sine of angle 'a' is calculated as the opposite side (3) over the hypotenuse (5), resulting in 3/5.
• 🔢 The cosine of angle 'a' is the adjacent side (4) over the hypotenuse (5), resulting in 4/5.
• 🔢 The tangent of angle 'a' is the opposite side (3) over the adjacent side (4), resulting in 3/4.
• 🔄 Cosecant is the reciprocal of sine, so if sine is 3/5, cosecant is 5/3.
• 🔄 Secant is the reciprocal of cosine, so if cosine is 4/5, secant is 5/4.
• 🔄 Cotangent is the reciprocal of tangent, so if tangent is 3/4, cotangent is 4/3.

Q & A

What are the six basic trigonometric functions?

The six basic trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot).

What is the mnemonic 'sohcahtoa' used for?

The mnemonic 'sohcahtoa' is used to help remember the equations for the sine, cosine, and tangent functions, where 'S' stands for sine, 'O' for opposite, 'H' for hypotenuse, 'C' for cosine, 'A' for adjacent, and 'T' for tangent.

How is the sine of an angle calculated in a right triangle?

The sine of an angle in a right triangle is calculated as the ratio of the length of the side opposite the angle to the length of the hypotenuse.

What is the formula for cosine in the context of the provided script?

The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.

How do you find the tangent of an angle in a right triangle?

The tangent of an angle is found by dividing the length of the side opposite the angle by the length of the adjacent side.

What is the hypotenuse in a right triangle?

The hypotenuse is the longest side of a right triangle, which is opposite the right angle.

How do you label the sides of a right triangle when finding trigonometric functions?

You label the hypotenuse with 'hyp', the opposite side with 'OPP', and the adjacent side with 'adj'.

What is the cosecant of an angle and how is it related to the sine function?

The cosecant of an angle is the reciprocal of the sine function, meaning if the sine is the ratio of the opposite side to the hypotenuse, the cosecant is the ratio of the hypotenuse to the opposite side.

How do you calculate the secant of an angle?

The secant of an angle is the reciprocal of the cosine function, which is the ratio of the hypotenuse to the adjacent side.

What is the cotangent of an angle and how is it found?

The cotangent of an angle is the reciprocal of the tangent function, which is the ratio of the adjacent side to the opposite side.

In the script, what is the length of the hypotenuse when finding the trigonometric functions for angle a?

In the script, the length of the hypotenuse is given as 5 units.

What are the lengths of the sides opposite and adjacent to angle a in the script's example?

In the script's example, the length of the side opposite angle a is 3 units, and the length of the adjacent side is 4 units.

Outlines

00:00

📚 Introduction to Basic Trigonometric Functions

This paragraph introduces the topic of the video, which is to explain the six basic trigonometric functions using the mnemonic 'sohcahtoa'. The speaker emphasizes the importance of memorizing these functions and suggests using the phrase to remember the equations for sine (opposite/hypotenuse), cosine (adjacent/hypotenuse), and tangent (opposite/adjacent). The paragraph sets the stage for a practical example that will follow.

📐 Labeling Triangle Sides for Trigonometric Functions

The speaker proceeds to demonstrate how to label the sides of a right-angled triangle in relation to a given angle 'a'. The hypotenuse is identified by its position opposite the 90-degree angle and is labeled 'hyp'. The 'opposite' side to angle 'a' is labeled 'OPP', and the 'adjacent' side is labeled 'adj'. This labeling process is crucial for calculating the trigonometric functions of angle 'a'.

🧮 Calculating Basic Trigonometric Functions

The paragraph details the calculation of the sine, cosine, and tangent of angle 'a' using the labeled sides of the triangle. The sine of 'a' is calculated as the ratio of the opposite side to the hypotenuse (3/5), the cosine as the ratio of the adjacent side to the hypotenuse (4/5), and the tangent as the ratio of the opposite side to the adjacent side (3/4). These calculations are fundamental to understanding the basic trigonometric functions.

🔄 Deriving Reciprocal Trigonometric Functions

Building upon the basic trigonometric functions, the speaker explains how to find the reciprocal functions: cosecant, secant, and cotangent. The cosecant is the reciprocal of the sine, the secant of the cosine, and the cotangent of the tangent. The paragraph provides the specific values for these functions based on the previously calculated sine, cosine, and tangent values, completing the explanation of the six basic trigonometric functions.

Keywords

💡Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. In the video, the focus is on the six basic trigonometric functions, which are sine, cosine, tangent, cosecant, secant, and cotangent. These functions are essential for understanding the relationships between the angles and sides of a right-angled triangle, and they are used to solve various problems in mathematics and physics.

💡SOHCAHTOA

SOHCAHTOA is a mnemonic device used to remember the definitions of the sine, cosine, and tangent functions. The video script uses this phrase to help viewers memorize the trigonometric ratios: 'S' stands for 'sine', which is the ratio of the opposite side to the hypotenuse; 'O' for 'opposite'; 'H' for 'hypotenuse'; 'C' for 'cosine', which is the ratio of the adjacent side to the hypotenuse; 'A' for 'adjacent'; 'T' for 'tangent', which is the ratio of the opposite side to the adjacent side; and 'OA' for the reciprocal of tangent, which is the ratio of the adjacent side to the opposite side.

💡Hypotenuse

The hypotenuse is the longest side of a right-angled triangle, which is opposite the right angle. In the context of the video, the hypotenuse is identified by drawing an arrow from the 90-degree angle to the opposite side. The hypotenuse is crucial for calculating trigonometric functions as it serves as the denominator in the sine and cosine ratios.

💡Opposite Side

The opposite side is the side of a right-angled triangle that is opposite the angle being considered. In the video, the opposite side is labeled with the length of three and is used in the calculation of sine and tangent functions. It is a key component in determining the values of these trigonometric ratios.

💡Adjacent Side

The adjacent side is the side of a right-angled triangle that is next to the angle being considered but not the hypotenuse. In the video, the adjacent side is labeled with the length of four and is used in the calculation of cosine and tangent functions. It is essential for finding the values of these trigonometric ratios.

💡Sine

Sine is a trigonometric function that represents the ratio of the length of the side opposite the angle to the length of the hypotenuse. In the video, the sine of angle 'a' is calculated as 3 over 5, using the lengths of the opposite and hypotenuse sides from the example triangle.

💡Cosine

Cosine is a trigonometric function that represents the ratio of the length of the adjacent side to the length of the hypotenuse. The video demonstrates how to calculate the cosine of angle 'a' as 4 over 5, using the lengths of the adjacent and hypotenuse sides.

💡Tangent

Tangent is a trigonometric function that represents the ratio of the length of the opposite side to the length of the adjacent side. The video script explains that the tangent of angle 'a' is calculated as 3 over 4, based on the given side lengths in the example triangle.

💡Cosecant

Cosecant is the reciprocal of the sine function, representing the ratio of the hypotenuse to the length of the opposite side. In the video, the cosecant of angle 'a' is found by flipping the sine ratio, resulting in 5 over 3, based on the sine value previously calculated.

💡Secant

Secant is the reciprocal of the cosine function, representing the ratio of the hypotenuse to the length of the adjacent side. The video shows how to find the secant of angle 'a' by flipping the cosine ratio, yielding 5 over 4, based on the cosine value previously calculated.

💡Cotangent

Cotangent is the reciprocal of the tangent function, representing the ratio of the adjacent side to the length of the opposite side. The video script illustrates that the cotangent of angle 'a' is found by flipping the tangent ratio, resulting in 4 over 3, based on the tangent value calculated earlier.

Highlights

Introduction to the six basic trigonometric functions and the mnemonic 'sohcahtoa' for memorization.

Explanation of how 'soh' in 'sohcahtoa' stands for 'sine', 'opposite', and 'hypotenuse'.

Demonstration of the 'cough' part of 'sohcahtoa' for remembering the cosine function formula.

Illustration of the 'Toa' mnemonic for the tangent function with 'opposite', 'adjacent'.

Step-by-step labeling of the sides of a right triangle for trigonometric calculations.

Identification of the hypotenuse as the longest side opposite the 90-degree angle.

Method to label the opposite side of the angle for which trigonometric functions are being found.

Description of how to label the adjacent side, touching the angle but not the hypotenuse.

Calculation of sine using the formula opposite/hypotenuse with given side lengths.

Calculation of cosine using the formula adjacent/hypotenuse with the given triangle.

Finding the tangent of angle 'a' using the formula opposite/adjacent.

Derivation of the cosecant function by inverting the sine ratio.

Explanation of finding the secant by inverting the cosine ratio.

Method to calculate the cotangent by inverting the tangent ratio.

Emphasis on the ease of finding the reciprocal trigonometric functions once the basic functions are known.

Practical application of the mnemonic 'sohcahtoa' in solving trigonometric problems.

Visual demonstration of labeling and calculating trigonometric functions for a given angle in a triangle.

Transcripts

so welcome to my video on the six basic

trig functions in this example I'm going

to find the value for the six basic trig

functions for this angle a and many

people have trouble memorizing the

equations for the basic trig functions

and one way that's easy to memorize the

equations is using the phrase so Chi Toa

I know it sounds really weird but it

helps out a lot

the phrase is sohcahtoa and if you look

at the equation for the sine you see

that the sine is equal to the opposite

over the hypotenuse and you can use the

word so to help you remember this

equation because the S stands for the

sine the O stands for the opposite and

the H stands for the hypotenuse and you

can do the same thing for the cosine in

the equation the cosine is equal to the

adjacent over the hypotenuse you can use

the word cough to help you remember that

the C stands for the cosine the a stands

for the adjacent and the H stands for

the hypotenuse and I think you can see

the pattern by now you can do the same

thing for the word Toa the T stands for

tangent the O stands for the opposite

and the a stands for the adjacent so

tangent is equal to opposite over

adjacent so let's get started right away

with this example we want to find the

values of the basic trig functions for

this angle a and the first thing that I

like to do is I like to label all the

sides of the triangle and the first side

that I always like to label is the

hypotenuse the hypotenuse is the longest

side and the easiest way to find the

hypotenuse is to go to the 90-degree

angle and if you draw an arrow to the

opposite side of the 90-degree angle

that side is always going to be your

hypotenuse so now I'm going to label the

hypotenuse with an hyp so our side with

five is the hypotenuse so now I'm going

to label the opposite side the opposite

side is always opposite of the angle

that we're trying to find so since we're

trying to find the trig functions for

angle a I'm going to go to angle a now

I'm going to draw an arrow to the

opposite side so our side with the

length of three is going to be the

opposite of angle a so now I'm going to

label side three with an OPP just to

show that it's the opposite side and

lastly we need to label the adjacent

side and the adjacent side is always

touching the angle but not the

hypotenuse so notice this side that I'm

calling over in red is touching our

angle a but it's not the hypotenuse so

that's our adjacent side the side with

the length of four is going to be our

adjacent side and I'm going to label the

side with an adj just to show that it's

the adjacent side and after we label all

the sides of the triangle it's really

easy to find the values for all the trig

functions for our angle a so that's what

I'm going to do right now first I'm

going to start off with the sine of a

and if we go back to our equation the

sine is equal to the opposite over the

hypotenuse so since our opposite is

equal to three and our hypotenuse is

equal to five the sine of a is equal to

3 over 5 so now I'll do the cosine of a

if we go back to our equation the cosine

is equal to the adjacent over the

hypotenuse since the adjacent has the

length of 4 and the hypotenuse has a

length of 5 the cosine of a is equal to

4 over 5 so now do the tangent of a we

go back to our equation the tangent is

equal to the opposite over the adjacent

since our opposite side has a length of

3 in our jacent side has a length of 4

the tangent of a is equal to 3 over 4

and once you have these

basic trig functions it's really easy to

find the last three basic trig functions

so now I'll do the cosecant of a and the

only thing you need to do to find the

cosecant of a is flip the sign of a

since the sine of a is three over five

the cosecant of a is five over three and

now I'm going to find the secant of a

and the only thing you need to do to

find a secant is flip the cosine since

the cosine of a is four over five the

secant of a is five over four and

finally we need to find the cotangent of

a in order to find the cotangent you

just need to flip the tangent so since

the tangent of a is three over four the

cotangent of a is four over three