Cofunction Identities and
Reflection
Bradley Hughes
Larry Ottman
Lori Jordan
Mara Landers
Andrea Hayes
Brenda Meery
Art Fortgang
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Printed: April 2, 2014
AUTHORS
Bradley Hughes
Larry Ottman
Lori Jordan
Mara Landers
Andrea Hayes
Brenda Meery
Art Fortgang
www.ck12.org
C HAPTER
Chapter 1. Cofunction Identities and Reflection
1
Cofunction Identities and
Reflection
Here you’ll learn about the four cofunction identities and how to apply them to solve for the values of trig functions.
While toying with a triangular puzzle piece, you start practicing your math skills to see what you can find out about it.
You realize one of the interior angles of the puzzle piece is 30◦ , and decide to compute the trig functions associated
with this angle. You immediately want to compute the cosine of the angle, but can only remember the values of your
sine functions.
Is there a way to use this knowledge of sine functions to help you in your computation of the cosine function for
30◦ ?
Read on, and by the end of this Concept, you’ll be able to apply knowledge of the sine function to help determine
the value of a cosine function.
Watch This
MEDIA
Click image to the left for more content.
Cofunctions
Guidance
In a right triangle, you can apply what are called "cofunction identities". These are called cofunction identities
because the functions have common values. These identities are summarized below.
sin θ = cos(90◦ − θ)
cos θ = sin(90◦ − θ)
tan θ = cot(90◦ − θ)
cot θ = tan(90◦ − θ)
Example A
Find the value of cos 120◦ .
Solution: Because this angle has a reference angle of 60◦ , the answer is cos 120◦ = − 12 .
Example B
Find the value of cos(−120◦ ).
Solution: Because this angle has a reference angle of 60◦ , the answer is cos(−120◦ ) = cos 240◦ = − 12 .
1
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Example C
Find the value of sin 135◦ .
Solution: Because this angle has a reference angle of 45◦ , the answer is sin 135◦ =
√
2
2
Vocabulary
Cofunction Identity: A cofunction identity is a relationship between one trig function of an angle and another trig
function of the complement of that angle.
Guided Practice
1. Find the value of sin 45◦ using a cofunction identity.
2. Find the value of cos 45◦ using a cofunction identity.
3. Find the value of cos 60◦ using a cofunction identity.
Solutions:
√
1. The sine of
is equal to
=
= 22 .
√
◦
◦
◦
◦
2. The cosine of 45 is equal to sin(90 − 45 ) = sin 45 = 2 2 .
45◦
cos(90◦ − 45◦ )
cos 45◦
3. The cosine of 60◦ is equal to sin(90◦ − 60◦ ) = sin 30◦ = .5.
Concept Problem Solution
Since you now know the cofunction relationships, you can use your knowledge of sine functions to help you with
the cosine computation:
√
cos 30◦ = sin(90◦ − 30◦ ) = sin(60◦ ) = 2 3
Practice
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
2
Find a value for θ for which sin θ = cos 15◦ is true.
Find a value for θ for which cos θ = sin 55◦ is true.
Find a value for θ for which tan θ = cot 80◦ is true.
Find a value for θ for which cot θ = tan 30◦ is true.
Use cofunction identities to help you write the expression tan 255◦ as the function of an acute angle of measure
less than 45◦ .
Use cofunction identities to help you write the expression sin 120◦ as the function of an acute angle of measure
less than 45◦ .
Use cofunction identities to help you write the expression cos 310◦ as the function of an acute angle of measure
less than 45◦ .
Use cofunction identities to help you write the expression cot 260◦ as the function of an acute angle of measure
less than 45◦ .
Use cofunction identities to help you write the expression cos 280◦ as the function of an acute angle of measure
less than 45◦ .
Use cofunction identities to help you write the expression tan 60◦ as the function of an acute angle of measure
less than 45◦ .
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Chapter 1. Cofunction Identities and Reflection
11. Use cofunction identities to help you write the expression sin 100◦ as the function of an acute angle of measure
less than 45◦ .
12. Use cofunction identities to help you write the expression cos 70◦ as the function of an acute angle of measure
less than 45◦ .
13. Use cofunction identities to help you write the expression cot 240◦ as the function of an acute angle of measure
less than 45◦ .
14. Use a right triangle to prove that sin θ = cos(90◦ − θ).
15. Use the sine and cosine cofunction identities to prove that tan(90◦ − θ) = cot θ.
3