# The Trigonometry of Right Triangles

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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 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 =

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 =

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 =

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

Trigonometry and Right Triangles visit the links below: Right Triangle Solvers The Six Functions

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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

Sorry, that’s not correct!

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

Sorry, that’s not correct!

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

Sorry, that’s not correct!