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Unit 34. TRIGONOMETRIC FUNCTIONS WITH RIGHT TRIANGLES. VARIATION OF FUNCTIONS. As the size of an angle increases , the sine , tangent , and secant functions increase , but the cofunctions (cosine, cotangent, cosecant) decrease Which is greater: cos 38° or cos 43°?

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unit 34

Unit 34

TRIGONOMETRIC FUNCTIONS WITH RIGHT TRIANGLES

variation of functions
VARIATION OF FUNCTIONS
  • As the size of an angle increases, the sine, tangent, and secant functions increase, but the cofunctions (cosine, cotangent, cosecant) decrease
  • Which is greater: cos 38° or cos 43°?
    • Since the cosine function decreases as the size of the angle increases, cos 38° is greater than cos 43°
  • Which is greater: tan 42° or tan 24°?
    • Since the tangent function increases as the size of the angle increases, tan 42° is greater than tan 24°
functions of complementary angles

sin A = cos (90° – A)

cos A = sin (90° – A)

tan A = cot (90° – A)

cot A = tan (90° – A)

sec A = csc (90° – A)

csc A = sec (90° – A)

FUNCTIONS OF COMPLEMENTARY ANGLES
  • Two angles are complementary when their sum is 90°. For example, 30° is the complement of 60°, and 60° is the complement of 30°
  • A function of an angle is equal to the cofunction of the complement of the angle
finding unknown angles
FINDING UNKNOWN ANGLES
  • Procedure for determining an unknown angle when two sides are given:
  • In relation to the desired angle, identify two given sides as adjacent, opposite, or hypotenuse
  • Determine the functions that are ratios of the sides identified in relation to the desired angle
  • Choose one of the two functions, substitute the given sides in the ratio
  • Determine the angle that corresponds to the quotient of the ratio
example of finding an angle

B

3.2"

5.7"

EXAMPLE OF FINDING AN ANGLE
  • Determine B to the nearest tenth of a degree in the triangle below:
example cont
EXAMPLE (Cont)
  • Since 5.7" is opposite B and 3.2" is adjacent B; either the tangent or the cotangent function could be used
  • Remember that when looking for an angle you will use arctan in this case or tan-1 on your calculator
  • Choosing the tangent function:
finding unknown sides
FINDING UNKNOWN SIDES
  • Procedure for determining an unknown side when an angle and a side are given:
  • In relation to the given angle, identify the given side and the unknown side as adjacent, opposite, or hypotenuse
  • Determine the trigonometric functions that are ratios of the sides identified in relation to the given angle
  • Choose one of the two functions and substitute the given side and given angle
  • Solve as a proportion for the unknown side
example of finding a side

x

46.3°

2.7 cm

EXAMPLE OF FINDING A SIDE
  • Determine side x (to the nearest hundredth) of the right triangle shown below:
example cont1
EXAMPLE (Cont)
  • In relation to the 46.3° angle, the 2.7 cm side is the adjacent side and side x is the hypotenuse. Thus, either the cosine or the secant function could be used
  • Choosing the cosine function:

so x = 3.91 cm Ans

practice problems
PRACTICE PROBLEMS
  • Which is greater: sin 48° or sin 32°?
  • Which is greater: csc 54.3° or csc 45.3°?
  • What is the cofunction of the complement of sec 35°?
  • What is the cofunction of the complement of cos 82°?
practice problems cont

A

4.4"

3.2"

PRACTICE PROBLEMS (Cont)
  • Determine angle A to the nearest tenth of a degree in the triangle shown below:
practice problems cont1

b

28°

5.4 mm

PRACTICE PROBLEMS (cont)
  • Determine side b in the triangle given below. Round your answer to two decimal places.
  • Determine 1 (to the nearest tenth of a degree) in the triangle given below.

5.86"

1

3.25"

problem answer key
PROBLEM ANSWER KEY
  • sin 48°
  • csc 45.3°
  • csc 55°
  • sin 8°
  • 46.7°
  • 4.77 mm
  • 33.7°