Cofunction Identities
Cofunction Identities: this Effortless Math guide explains the topic in plain language, shows how it works through solved examples, and gives you free practice to try immediately, so the idea sticks instead of staying abstract.
Cofunction identities show the relationship between the different trigonometric functions and their complementary angles. In this guide, you will learn more about cofunction identities.
Cofunction Identities: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Choose the modelUse a right triangle, the unit circle, or a transformed graph.
- Track unitsConvert degrees and radians when needed.
- Use identitiesReplace complicated trig expressions with equivalent simpler ones.
Worked examples
Right-triangle sine
- Sine is opposite over hypotenuse.
- Substitute 5 and 13.
- Leave the ratio simplified.
Unit-circle cosine
- At angle 0, the point is (1, 0).
- Cosine is the x-coordinate.
- Read the x-value.
Try one before moving on
Cofunction Identities: pop-up practice
A step-by-step guide to cofunction identities
Cofunction identities are trigonometric identities that show a relationship between trigonometric functions and complementary angles.
We have six identities that can be obtained using right triangles, the angle sum property of a triangle, and trigonometric ratio formulas.
The cofunction identities establish a relationship between trigonometric functions (sin) and (cos), (tan) and (cot), and (sec) and (csc). These functions are known as cofunctions of each other.
We can write cofunction identities in terms of radians and degrees because these are the units of angle measurement.
Cofunction identities in radians
- (color{blue}{sin:left(frac{pi }{2}:-:θright)=cos:θ})
- (color{blue}{cos:left(frac{pi }{2}:-:θright)=sin:θ})
- (color{blue}{tan:left(frac{pi }{2}-:θright)=cot:θ})
- (color{blue}{cot:left(:frac{pi }{2}-θright)=tan:θ})
- (color{blue}{sec:left(frac{pi }{2}-:θright)=cosec:θ})
- (color{blue}{csc:left(frac{pi }{2}-θright)=sec:θ})
Cofunction identities in degrees
- (color{blue}{sin:left(90°:-:θright)=cos:θ})
- (color{blue}{cos:left(90°:-:θright)=sin:θ})
- (color{blue}{tan:left(90°:-:θright)=cot:θ})
- (color{blue}{cot:left(90°:-:θright)=tan:θ})
- (color{blue}{sec:left(90°:-:θright)=cosec:θ})
- (color{blue}{csc:left(90°-:θright)=sec:θ})
Cofunction Identities, Example 1:
Find the value of acute angle (x), if (sin:x=cos:40°).
Solution:
Using cofunction identity, (cos:left(90°:-:θright)=sin:θ), we can write (sin:x=cos:40°) as:
(sin:x=cos:40°)
(cos:left(90°-:xright)=cos:40°)
(90°-:x=40°)
(x=90°-40°)
(x=50°)
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