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Cosine
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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.
CB
A
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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).
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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)
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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)
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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.
Hypotenuse
CB
A
Adjacent
Opposite
Hypotenuse
CB
A
Adjacent
Opposite
With respect to angle B With respect to angle A
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Sine - SOH
When asked to determine the sine of an angle, for example sin(A):
1) Identify the side opposite the angle
2) Identify the hypotenuse
3) Substitute those values into the equation
SOH Sin(θ) = Opp Hyp
Opposite
Hypotenuse
3 CB
A
460°5
Example
sin(60) = 3 = 0.60 5
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Cosine - CAH
When asked to determine the cosine of an angle, for example cos(A):
1) Identify the side opposite the angle
2) Identify the hypotenuse
3) Substitute those values into the equation
CAH Cos(θ) = Adj Hyp
3
45
Example
cos(60) = 4 = 0.80 5
AdjacentHypotenuse
CB
A
60°
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Tangent - TOA
When asked to determine the tangent of an angle, for example tan(A):
1) Identify the side opposite the angle
2) Identify the hypotenuse
3) Substitute those values into the equation
TOA Tan(θ) = Opp Adj
Opposite3
A
CB
60° Adjacent4
5
Example
tan(60) = 3 = 0.75 4
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SOH- CAH- TOA
Just Remember…
SOH Sin(θ) = Opp/Hyp
CAH Cos(θ) = Adj/Hyp
TOA Tan(θ) = Opp/Adj
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Additional Resources
To learn more about Trigonometry and Right Triangles
visit the links below:
Trigonometry and Right Triangles
Right Triangle Solvers
The Six Functions
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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.
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C B
A
Practice
Click on the side opposite of angle B.
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Practice
Click on the side opposite of angle B.
That’s Correct!
C B
A
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Practice
Click on the side opposite of angle B.
Try Again
C B
A
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Practice
Click on the side adjacent to angle B.
CB
A
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Practice
Click on the side adjacent to angle B.
CB
A
That’s Correct!
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Practice
Click on the side adjacent to angle B.
CB
A
Try Again
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Practice: Sine
CB
A
8
What is sin(62°)?
1517
62°
a) 15/17
b) 8/17
c) 17/15
d) 8/15
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That’s Correct!
Click here to continue.
sin(62°) = 15/17
CB
A
8
1517
62°
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Sorry, that’s not correct!
Click here to continue.
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Practice: Cosine
What is cos(62°)?
CB
A
8
1517
62°
a) 15/17
b) 8/17
c) 17/15
d) 8/15
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That’s Correct!
Click here to continue.
cos(62°) = 8/17
CB
A
8
1517
62°
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Sorry, that’s not correct!
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Practice: Tangent
What is tan(62°)?
a) 15/17
b) 8/17
c) 17/8
d) 15/8CB
A
8
1517
62°
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That’s Correct!
Click here to continue.
tan(62°) = 8/15
CB
A
8
1517
62°
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Sorry, that’s not correct!
Click here to continue.
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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
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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
