Pc Trig Applications

Trigonometric Applications and Models

Trigonometric Functions on a Calculator Example 1: Calculate sin 40 . Set the calculator in degree mode. Calculator keystrokes: sin 40 = Display: 0.6427876 Example 2: Calculate sec 40 . Calculator keystrokes: 1 cos 40 = Display: 1.3054072

Solving Right Triangles Solving a right triangle means to find the lengths of the sides and the measures of the angles of a right triangle. Some information is usually given. a

θ

a θ

• an angle θ and a side a, b a

• or two sides, a and b.

a

θ

a

b

a b

Solving A Right Triangle Given an Angle and a Side Solve the right triangle. The third angle is 60 , the complement 30○ of 30 . Use the values of the trigonometric functions of 30o.

5

1 op Since = sin 30 = = 5 , it follows that hyp = 10. phyp hyp 2 To get the last side, note that 10 60○ 5 3 adj ; 30○ = cos 30 = 2 10 5 3 therefore, adj = 5 3

Example 1: A bridge is to be constructed across a small river from post A to post B. A surveyor walks 100 feet due south of post A. She sights on both posts from this location and finds that the angle between the posts is 73 . Find the distance across the river from post A to post B. x Post B Post A Use a calculator to find tan 73o = 3.27. 100 ft. ○ x opp 73 3.27 = tan 73 = = 100 adj It follows that x = 327. The distance across the river from post A to post B is 327 feet.

Inverse Trigonometric Functions on a Calculator Labels for sin−1, cos−1, and tan−1 are usually written above the sin, cos, and tan keys. Inverse functions are often accessed by using a key that maybe be labeled SHIFT, INV, or 2nd. Check the manual for your calculator. Example: Find the acute angle θ for which cos θ = 0.25. Calculator keystrokes: (SHIFT) cos−1 0.25 = Display: 75.22487

Solving a Right Triangle Given Two Sides Solve the right triangle shown.

5

Solve for the hypotenuse: hyp2 = 62 + 52 = 61 hyp = 61 = 7.8102496

θ 6 61

Solve for θ :

5 opp 5 -1 tan θ = = and θ = tan ( ). 6 6 adj

Calculator Keystrokes: (SHIFT) tan−1 ( 5 Display: 39.805571

39.8○ 6 6)

Subtract to calculate the third angle: 90 − 39.805571 = 50.194428 .

50.2○

5

Angle of Elevation and Angle of Depression When an observer is looking upward, the angle formed by a horizontal line and the line of sight is called the: angle of elevation.

line of sight object angle of elevation horizontal

observer

When an observer is looking downward, the angle formed by a horizontal line and the line of sight is called the: angle of depression.

horizontal angle of depression line of sight object

observer

Example 2: A ship at sea is sighted by an observer at the edge of a cliff 42 m high. The angle of depression to the ship is 16 . What is the distance from the ship to the base of the cliff? observer cliff 42 m

horizontal 16○ angle of depression line of sight 16○ d

d=

42 = 146.47. tan 16°

The ship is 146 m from the base of the cliff.

ship

Example 3: A house painter plans to use a 16 foot ladder to reach a spot 14 feet up on the side of a house. A warning sticker on the ladder says it cannot be used safely at more than a 60 angle of inclination. Does the painter’s plan satisfy the safety requirements for the use of the ladder? ladder house 14 16 sin θ = = 0.875 14 16 θ Next use the inverse sine function to find θ.

θ = sin−1(0.875) = 61.044975 The angle formed by the ladder and the ground is about 61 . The painter’s plan is unsafe!