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

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).

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

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 0.5. This statement is written as follows: 0.5 = cos(60)

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

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 = 0.60 5

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 = 0.80 5

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 = 0.75 4

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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