Unit 34

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# Unit 34 - PowerPoint PPT Presentation

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

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°?
• 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°

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
• 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

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)
• 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
• 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

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 (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
• 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°?

A

4.4"

3.2"

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

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"