MTM 2 Notes
Unit 58
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Unit 58: Introduction to Trigonometric Functions
Similar triangles have side lengths that are the same proportion no matter how big the triangles are.
Notice that all the angles are the same for similar triangles. Thus, for an angle of 36.86990 in a right triangle,
the ratio of the opposite side length to the adjacent side length will always be 0.75. This can be used, for
example, to find the length of a side of a right triangle where the side adjacent to an angle of 36.86990 is 80
feet.
Practice:
1. Find the length of the leg opposite a 36.86990 angle in a right triangle when the adjacent leg is 80 feet
long.
Mathematicians have figured out functions that compute the above three ratios (plus 3 additional ratios) given
just the angle. These six functions are called the sine, cosine, tangent, cotangent, secant, and cosecant
functions. These six functions are always equal to the ratios in a right triangle shown in the table on the next
page. These functions are important because we can use them to find missing sides of right triangles just like in
the previous example.
MTM 2 Notes
Unit 58
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The Ratios of the Six Trigonometric Functions:
Function
Sine of Angle A
Symbol
sin(A)
Cosine of Angle A
cos(A)
Tangent of Angle A
tan(A)
Cotangent of Angle A
cot(A)
Secant of Angle A
sec(A)
Cosecant of Angle A
csc(A)
Practice: Round each answer to six decimal places.
2. Determine the sine, cosine, and tangent of 12.
3. Determine the sine, cosine, and tangent of 45.
4. Determine the sine, cosine, and tangent of 87.95.
5. Determine the sine, cosine, and tangent of 52 23 47.
Function Definition
opp a
sin  A 

hyp c
adj b
cos  A 

hyp c
opp a
tan  A 

adj b
adj b
cot  A 

opp a
hyp c
sec  A 

adj b
hyp c
csc  A 

opp a
MTM 2 Notes
Unit 58
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To find the Cotangent, Secant, and Cosecant Functions:
These functions are never built into any calculators. To calculate these functions, you have to use the fact that
these three functions are reciprocals of the sine, cosine, and tangent functions.
cot  A 
1
tan  A
sec  A 
1
cos  A
Practice: Round each answer to six decimal places.
6. Determine the cotangent, secant, and cosecant of 12.
7. Determine the cotangent, secant, and cosecant of 45.
8. Determine the cotangent, secant, and cosecant of 87.95.
9. Determine the cotangent, secant, and cosecant of 52 23 47.
csc  A 
1
sin  A
MTM 2 Notes
Unit 58
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To find an angle given a ratio of sides:
Imagine you have a right triangle where you know the lengths of the sides, which means you can compute ratios
of the sides, but you do not know the angles. Mathematicians have devised functions called the inverse
trigonometric functions (or arc functions) that can tell you the angle of the triangle if you know a value for the
appropriate ratio. The buttons for these functions are usually labeled sin-1 for inverse sine, cos-1 for inverse
cosine, and tan-1 for inverse tangent.
For example, in the triangle below, you might want to know what Angle A is equal to:
Practice:
10. Compute the length of the hypotenuse c.
11. Write an equation for Angle A using the sine function.
12. Compute Angle A using the inverse sine function.
13. Write an equation for Angle A using the cosine function.
14. Compute Angle A using the inverse cosine function.
15. Write an equation for Angle A using the tangent function.
16. Compute Angle A using the inverse tangent function.
MTM 2 Notes
Unit 58
17. Compute Angle A: sin(A) = 0.59716
18. Compute Angle A: cos(A) = 0.12895
19. Compute Angle A: tan(A) = 0.33641
To find an angle given a value for Cotangent, Secant, or Cosecant:
Use the reciprocal identities…
1
 If cot  A  x , then A  tan 1   .
 x
1
 If sec  A  x , then A  cos 1   .
 x
1
 If csc  A  x , then A  sin 1   .
 x
Practice:
20. Compute Angle A: cot(A) = 0.12345
21. Compute Angle A: cot(A) = 0.58914
22. Compute Angle A: sec(A) = 2.5981
23. Compute Angle A: csc(A) = 5.6987
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