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Trigonometry. Basic Calculations of Angles and Sides of Right Triangles. Introduction. You can use the three trig functions ( sin , cos , and tan ) to solve problems involving right triangles. Using a Calculator. 74. or. 74. 74. or. 74. 74. or. 74. cos. sin. sin. tan. tan.

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trigonometry

Trigonometry

Basic Calculations of Angles and Sides of Right Triangles

introduction
Introduction
  • You can use the three trig functions (sin, cos, andtan) to solve problems involving right triangles.
slide3

Using a Calculator

74

or

74

74

or

74

74

or

74

cos

sin

sin

tan

tan

cos

0.961261695

0.275637355

3.487414444

Enter

Enter

Enter

You can use a calculator to approximate the sine, the cosine, and the tangent of 74º. Make sure your calculator is in degree mode. The table shows some sample keystroke sequences accepted by most calculators.

0.9613

0.2756

3.4874

slide4

Find a missing side length

Use trigonometry to determine the length of a side of a right triangle.

determining the length of a side example 5
Determining the length of a sideExample 5
  • In this problem, we will determine the length of side x.

9”

x

26°

determining the length of a side example 51
Determining the length of a sideExample 5
  • As always, first label the sides of the triangle...

hypotenuse

9”

opposite

x

26°

adjacent

determining the length of a side example 52
Determining the length of a sideExample 5
  • Since you know the length of the hypotenuse and want to know the length of the opposite side, you should pick a trig function that contains both of them...

hypotenuse

9”

opposite

x

26°

determining the length of a side example 53
Determining the length of a sideExample 5
  • Which trig function should you pick?

You need to pick the sine function since it is the only one that has both the opposite side and hypotenuse in it.

hypotenuse

9”

x

opposite

26°

determining the length of a side example 54

Use basic algebra to solve this equation. Multiply both sides of the equation by 9 to clear the fraction.

Determining the length of a sideExample 5
  • Now set-up the trig function:

hypotenuse

9”

opposite

x

26°

determining the length of a side example 55
Determining the length of a sideExample 5
  • Now you know the opposite side has a length of 3.95”.

hypotenuse

9”

opposite

3.95”

26°

determining the length of a side example 6

x

75 mm

47°

Determining the length of a sideExample 6
  • Let’s try another one.
  • Determine the length of side x.
determining the length of a side example 61

x

75 mm

47°

Determining the length of a sideExample 6
  • Since the known angle (47°) will serve as your reference angle, you can label the sides of the triangle...

opposite

adjacent

hypotenuse

determining the length of a side example 62

x

75 mm

47°

Determining the length of a sideExample 6
  • You know the length of the hypotenuse and want to know the length of the adjacent side, so pick a trig function which contains both of them...

adjacent

hypotenuse

determining the length of a side example 63

x

75 mm

47°

Determining the length of a sideExample 6
  • Which trig function should you pick?

You need to pick the cosine function since it is the only one that has both the adjacent side and hypotenuse in it.

adjacent

hypotenuse

determining the length of a side example 64

Use basic algebra to solve this equation. Multiply both sides of the equation by 75 to clear the fraction.

To finish, evaluate cos 47° (which is 0.682) and multiply by 75.

x

75 mm

47°

Determining the length of a sideExample 6
  • Set-up your trig function...

adjacent

hypotenuse

determining the length of a side example 65
Determining the length of a sideExample 6
  • Now you know the length of the adjacent side is 51.1 mm.

51.1 mm

75 mm

adjacent

hypotenuse

47°

determining the length of a side example 7
Determining the length of a sideExample 7
  • Let’s try a little bit more challenging problem.
  • Determine the length of side x.

x

12 ft

35°

determining the length of a side example 71
Determining the length of a sideExample 7
  • Label the sides of the right triangle...

hypotenuse

x

opposite

12 ft

35°

adjacent

determining the length of a side example 72
Determining the length of a sideExample 7
  • Which trig function will you pick? You know the length of the side opposite and want to know the length of the hypotenuse.

hypotenuse

x

opposite

12 ft

35°

adjacent

determining the length of a side example 73
Determining the length of a sideExample 7
  • Which trig function should you pick?

You need to pick the sine function since it is the only one that has both the opposite side and hypotenuse in it.

hypotenuse

opposite

x

12 ft

35°

determining the length of a side example 74

Use algebra to solve this equation. Multiply both sides of the equation by x to clear the fraction.

Next, divide both sides by sin35° to isolate the unknown x.

Determining the length of a sideExample 7
  • Now set-up your trig function.

hypotenuse

x

opposite

12 ft

35°

determining the length of a side example 8

35 cm

50°

x

Determining the length of a sideExample 8
  • The reason the last problem was a little bit more difficult was the fact that you had an unknown in the denominator of the fraction.
  • Keep clicking to see a similar trig function solved.
determining the length of a side example 9

65°

45.5 mm

y

x

Determining the length of a sideExample 9
  • Your turn.
  • Determine the lengths of sides xandy.
slide24

65°

45.5 mm

y

x

Determining the length of a sideExample 9

  • Since you want to know the length of side y (adjacent) and you know the length of the hypotenuse, which trig function should you select?

hypotenuse

adjacent

opposite

slide25

65°

45.5 mm

19.2 mm

25°

41.3 mm

Determining the length of a sideExample 9

  • Both sides have been determined, one by trig, the other using the Pythagorean Theorem.
  • Also the size of the other acute interior angle is indicated...
slide26

60°

x

7.5”

Summary

  • After viewing this lesson you should be able to:
    • Compute the length of any side of a right triangle as long as you know the length of one side and an acute interior angle.