1. The Trigonometry of Right Triangles
This tutorial will teach you how to solve the three primary trigonometric functions using:
1) right triangles, and
2) the mnemonic, SOH–CAH-TOA. C B A

2. Instructional Overview
Learner Audience:
This tutorial is intended for high school or college Trigonometry students. This topic is often touched on in Algebra II, and it’s also applied in Calculus and Physics courses.
Learning objectives:
When given one of three primary trigonometric functions, students will be be able to identify it’s components.
Given a right triangle, with the side and angle measures present, students will correctly used the mnemonic SOH-CAH-TOA to solve three trigonometric functions:
- sin(θ)
cos(θ)
tan(θ).

3. Trigonometric Functions
Functions, f(x), are used as a way to associate a unique output for each input of a specified type.
This tutorial presents the three primary trigonometric functions:
sine sin(x)
cosine written as cos(x)
tangent tan(x) C B A

4. Trigonometric Functions
In trigonometry, the input value, x, is usually an angle, θ.
For cosine, when the input value is 60 degrees, the output value is This statement is written as follows:
0.5 = cos(60)

5. SOH- CAH- TOA The hypotenuse is always across from the 90° angle.
To use SOH- CAH- TOA, you must determine what sides are opposite and adjacent to the angle being input into the trigonometric functions.
With respect to angle B
With respect to angle A
Hypotenuse C B A
Adjacent
Opposite
Hypotenuse C B A
Adjacent
Opposite

6. Sine - SOH
When asked to determine the sine of an angle, for example sin(A):
Identify the side opposite the angle
Identify the hypotenuse
3) Substitute those values into the equation
SOH  Sin(θ) = Opp Hyp
Hypotenuse 3 C B A 4
60° 5
Opposite
Example
sin(60) = 3 =

7. Cosine - CAH
When asked to determine the cosine of an angle, for example cos(A):
Identify the side opposite the angle
Identify the hypotenuse
3) Substitute those values into the equation
CAH  Cos(θ) = Adj Hyp C B A
60°
Hypotenuse
Adjacent 5 4 3
Example
cos(60) = 4 =

8. Tangent - TOA
When asked to determine the tangent of an angle, for example tan(A):
Identify the side opposite the angle
Identify the hypotenuse
3) Substitute those values into the equation
TOA  Tan(θ) = Opp Adj
Opposite 3 A C B
60°
Adjacent 4 5
Example
tan(60) = 3 =

9. SOH- CAH- TOA SOH  Sin(θ) = Opp/Hyp CAH  Cos(θ) = Adj/Hyp
Just Remember…
SOH  Sin(θ) = Opp/Hyp
CAH  Cos(θ) = Adj/Hyp
TOA  Tan(θ) = Opp/Adj

10. Additional Resources To learn more about
Trigonometry and Right Triangles
visit the links below:
Right Triangle Solvers
The Six Functions

11. Copyright Copyright 2007 Sharisse Turnbull
Permission to copy this tutorial at no cost is granted to all teachers and students of non-profit schools.
Permission is also granted to all teachers and students of non-profit schools to make revisions to this tutorial for their own purposes, on the condition that this copyright page and the credits page remain part of the tutorial. Teachers and students who adapt the tutorial should add their names and affiliations to the credits page without deleting any names already there.

12. Practice
Click on the side opposite of angle B. C B A

13. Practice
Click on the side opposite of angle B. C B A
That’s Correct!

14. Practice
Click on the side opposite of angle B. C B A
Try Again

15. Practice
Click on the side adjacent to angle B. C B A

16. Practice
Click on the side adjacent to angle B. A B C
That’s Correct!

17. Practice
Click on the side adjacent to angle B. C B A
Try Again

18. Practice: Sine What is sin(62°)? a) 15/17 b) 8/17 c) 17/15 d) 8/15 17

19. That’s Correct! sin(62°) = 15/17 Click here to continue. 17 15 62° 8 AB A 17 15
62° 8
Click here to continue.

20. Sorry, that’s not correct!
Click here to continue.

22. Practice: Cosine What is cos(62°)? a) 15/17 b) 8/17 c) 17/15 d) 8/15

23. That’s Correct! cos(62°) = 8/17 Click here to continue. 17 15 62° 8 AB A 17 15
62° 8
Click here to continue.

24. Sorry, that’s not correct!
Click here to continue.

26. Practice: Tangent What is tan(62°)? a) 15/17 b) 8/17 c) 17/8 d) 15/8

27. That’s Correct! tan(62°) = 8/15 Click here to continue. 17 15 62° 8 AB A 17 15
62° 8
Click here to continue.

28. Sorry, that’s not correct!
Click here to continue.

29. Further Practice
Try these on your own and click continue to check your answers. Give your answer in fraction and decimal form. Round your answers to the nearest hundredth.
1) sin (37°)
2) tan (53°)
3) cos (37°)
4) cos (53°)
5) tan (37°)
37°
53° 20 15 25

30. Further Practice: Answers
How did you do?
Review the areas that you missed, and keep up the good work!!!
1) sin (37°) = 15/25 = 0.60
2) tan (53°) = 20/15 = 1.33
3) cos (37°) = 20/25 = 0.80
4) cos (53°) = 15/25 = 0.60
5) tan (37°) = 15/20 = 0.75
37°
53° 20 15 25