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Name: ___________________________________

Pre-calculus Notes: Chapter 5 – The Trigonometric Functions

Section 1 – Angles and Degree Measure

Use the word bank below to fill in the blanks below. You may use each term only once.

degree vertex negative angle terminal side positive angle initial side standard position

An angle may be generated by rotating one of two rays that share a

fixed endpoint known as a _____________________. One of the rays

is fixed to form the ____________________________ of the angle, and

the second ray rotates to form the ______________________________.

If the rotation is in a counterclockwise direction, the angle formed is a _____________________

If the rotation is clockwise, it is a ______________________________.

An angle with its vertex at the origin and its initial side along the positive x-axis is said to be in

_________________________________. In the figures below, all the angles are in standard position.

The most common unit used to measure angles is the _________________________. In order to obtain

a more accurate angle measure, Babylonians also measured angles in minutes and seconds. A degree

is subdivided into 60 equal parts known as minutes (1’), and the minute is subdivided into 60 equal

parts know as seconds (1’’).

Example 1

Change the angle measure to degrees, minutes, and seconds.

a. 15.735o b. 329.125o

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Example 2

Rewrite as a decimal rounded to the nearest thousandth.

a. 39o 5’ 34’’ b. 35o 12’ 7’’

If the terminal side of an angle that is in standard position coincides with one of the axes, the angle is

called a quadrantal angle, as in the figures below.

A full rotation around a circle is 3600. Measures of more than 3600 represent multiple rotations.

Example 3

Give the angle measure represented by each rotation.

a. 9.5 rotations clockwise b. 6.75 rotations counterclockwise

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Examine the table below to determine the definitions for coterminal and reference angles. It may

help to sketch each angle.

Coterminal Angles Reference Angles

Examples for 60o Non-Examples for 60o Example for 60o Non-Examples for 60o

420o

780o

-300o

-660o

120o

-60o

30o

60o

60o 120o

420o

780o

-300o

Examples for 135o Non-Examples for 135o Example for 135o Non-Examples for 135o

495o

855o

-225o

-585o

135o

45o

-135o

225o

45o -135o

225o

-225o

-45o

Examples for 210o Non-Examples for 210o Example for 210o Non-Examples for 210o

570o

930o

-150o

-510o

30o

210o

150o

-210o

30o -150o

150o

210o

-210o

Examples for -20o Non-Examples for -20o Example for -20o Non-Examples for -20o

340o

700o

-380o

-740o

20o

-340o

-20o

340o

20o -20o

340o

-340o

Coterminal Angles: ________________________________________________________________________

Reference Angles: _________________________________________________________________________

Example 4

Identify all angles that are coterminal with each angle. Then find one positive angle and one

negative angle that are coterminal with the angle.

a. 86o b. 294o

Example 5

If each angle is in standard position, determine a coterminal angle that is between 0o and 360o. State

the quadrant in which the terminal side lies.

a. 595o b. -777o

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Example 6

Find the measure of the reference angle for each angle.

a. 1200 b. -1350

c. 3120 d. -1950

Section 2 – Trigonometric Ratios in Right Triangles

Example 1

Find the values of sine, cosine, and tangent for A .

Example 2

a. If 5

6sec , find cos . b. If 8.0sin , find csc .

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Example 3

Find the values of the six trigonometric ratios for E .

Section 3 – Trigonometric Functions on the Unit Circle

xcos ysin x

ytan

x

1sec

y

1csc

y

xcot

Be cautious: division by zero is undefined, so there are

values of tangent, cotangent, secant, and cosecant that are

undefined.

Example 1

Use the unit circle to find each value.

a. sin(-900) b. cot 2700 c. sec 900 d. cos(-1800)

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Example 2

Use the unit circle to find the values of the six trigonometric functions for a 2100 angle.

Example 3

Find the values of the six trigonometric functions for angle in standard position if a point with the

coordinates (-15, 20) lies on its terminal side.

Example 4

Suppose is an angle in standard position whose terminal side lies in Quadrant III. If 5

4sin ,

find the values of the remaining five trigonometric functions of .

Example 5

Suppose is an angle in standard position whose terminal side lies in Quadrant IV. If 5

29sec ,

find the values of the remaining five trigonometric functions of .

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Section 4 – Applying Trigonometric Functions

Example 1

If J = 500 and j = 12, find r.

Example 2

The chair lift at a ski resort rises at an angle of 20.750 and attains a vertical height of 1200 feet.

a. How far does the chair lift travel up the side of the mountain?

b. A film crew in a helicopter records an overhead view of a skier’s downhill run from where she

gets off the chair lift at the top to where she gets back on the chair lift for her next run. If the

helicopter follows a level flight path, what is the length of that path?

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Example 3

A regular hexagon is inscribed in a circle with diameter 26.6 centimeters. Find the apothem of the

hexagon. (Apothem = the measure of the line segment from the center of the polygon to the midpoint

of one of its sides)

Angle of Elevation ________________________________________________________________________

Angle of Depression _______________________________________________________________________

Example 4

An observer in the top of a lighthouse determines that the angles of depression to two sailboats

directly in line with the lighthouse are 3.50 and 5.750, If the observer is 125 feet above sea level, find

the distance between the boats.

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Section 5.5 – Solving Right Triangles

Example 1

Solve each equation.

a. tan x = 1 b. 2

1sin x

Example 2

Evaluate each expression. Assume that all angles are in Quadrant I.

a.

5

2arccoscos b.

5

4costan 1 c.

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2arcsincos

Example 3

If g = 28 and h = 21, find H.

Example 4

Many cities place restrictions on the height and placement of skyscrapers in order to protect residents

from completely shaded streets. If a 100-foot building casts an 88-foot shadow, what is the angle of

elevation of the sun?

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Example 5

Solve each triangle described, given the triangle below. Round to the nearest tenth.

a. K = 400, k = 26 b. j = 65, l = 55

Section 5.6 – The Law of Sines

Example 1

Solve LMN if L = 290, M = 1120, and l = 22.

Example 2

A person in a hot-air balloon observes that the angle of depression to a building on the ground is

65.80. After ascending vertically 500 feet, the person now observes that the angle of depression is

70.20. How far is the balloonist now from the building?

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Example 3

Find the area of ABC if a = 4.7, c = 12.4, and B = 47020’.

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Section 7 – The Ambiguous Case for the Law of Sines

Example 1

Determine the number of possible solutions for each triangle.

a. A = 300, a = 8, b = 10 b. b = 8, c = 10, B = 1180

c. A = 630, a = 18, b = 25 d. A = 1050, a = 73, b = 55

Example 2

Find all solutions for each triangle. If no solutions exist, write ‘none’.

a. A = 980, a = 39, b = 22

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b. A = 72.20, a = 21, b = 22

Example 3

A group of contractors is constructing a 24-foot slide on a playground. The slide inclines 450 from the

horizontal. The access ladder measures 18 feet long. At what angle to the horizontal should the

contractors build the ladder?

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Section 8 – The Law of Cosines

Example 1

Suppose you want to fence a triangular lot. If two sides measure 84 feet and 78 feet and the angle

between the two sides is 1020, what is the length of the fence to the nearest foot?

Example 2

Solve each triangle.

a. A = 39.40, b = 12, c = 14 b. a = 19, b = 24.3, c = 21.8

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Example 3

Find the area of ABC if a = 24, b = 52, and c = 39.

Example 4

Find the area of ABC . Round to the nearest tenth.