Chapter 5 The Trigonometric Functions
5.1 Angles and Degree Measure
Definitions
 Angle – generated by rotating one of two rays that share a fixed ________________________
o Three key parts of an angle



o Ex:

Standard Position – an angle with its vertex at the _______________ and its initial side along the
__________________ x-axis.
o Positive angle –
o Negative angle –
o Ex:

Degree – the most common unit in angle measurement
o The degree is subdivided into __________ equal parts known as _________________________
o The _________________ is subdivided into _______ equal parts known as _________________
o Ex:

Quadrant angle – when the terminal side of an angle that is in standard position coincides with one of
the ______________.
o Ex:

Coterminal Angles – two angles in standard position that have the same ________________________.
o Cannot be same angle
o Angles differ in degree measure by multiples of ___________
o Each angle has ___________________________ coterminal angles
o Ex:

Reference Angle – if 𝛼 is a nonquadrantal angle in standard position, its reference angle is defined as
the acute angle formed by the terminal side of the given angle and the x-axis.
o Ex:
Ex 1: Longitude and Latitude can be expressed in degrees as a decimal value or in degrees, minutes and
seconds.
a. Change north latitude 15.735° to degrees, minutes, and seconds
b. Write north latitude 39° 5’ 34’’ as a decimal rounded to the nearest thousandth
Ex 2: Give the angle measure represented by each rotation.
a. 5.5 rotations clockwise
b. 3.3 rotations counterclockwise
Ex 3: 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. 45°
b. 225°
Ex 4: If each angle is in standard position, determine a coterminal angle that is between 0° and 360°. State
the quadrant in which the terminal side lies.
a. 775°
b. − 1297°
Ex 5: Find the measure of the reference angle for each angle
a. 312°
b. −135°
5.2 Trigonometric Ratios in Right Angles
Trigonometric Ratios
A ratio of the sides of a right triangle used to define the __________, ______________ and _______________
ratios of the triangle.
Ex 1: Write the ratios of the 6 trigonometric functions of ∠𝐴.
Ex 2: Let 𝑡𝑎𝑛𝑀 =
√5
.
4
What is 𝑠𝑖𝑛𝑀 and 𝑠𝑒𝑐𝑀?
5.4 Applying Trigonometric Functions
Ex 1: Find x and y to the nearest tenth.
Ex 2: Find the area of a regular dodecagon whose radius is 15cm. Round to the nearest thousandth.
Angle of Elevation vs. Angle of Depression
Ex 3: The angle of depression from Tim to Drake is 25° whereas the angle of depression from Tim to John is
15°. Find the distance between Drake and John to the nearest tenth of a foot.
5.5 Solving Right Triangles
Finding an Angle Measure
 Sometimes you know a trigonometric value of an angle but you do NOT know the measure of the
angle.
o To find the angle measure, you must use the ___________________ of the trig function
o The __________________ of the sine function is also called the ___________________
o Abbreviated:
Difference between Inverse and Reciprocals
Ex 1: Find the value of x. Round to the nearest hundredth.
Ex 2: Solve the right triangle. In other words, find ALL missing side lengths and angle measures.
Ex 3: Evaluating expressions without the use of a calculator.
1
a. sin(𝑡𝑎𝑛−1 (2))
4
b. cos(𝑐𝑠𝑐 −1 (3))
10
c. cot(𝑠𝑒𝑐 −1 ( 7 )
Ex 4: The High Roller Ferris Wheel in Vegas stands 550ft tall, with a diameter of 520 ft.
a. Find the height after 127° rotation (clockwise).
b. Find the height after 275° rotation (clockwise).
c. The height is 40 ft. How many degrees did it rotate (clockwise)?
5.3 Trigonometric Functions on the Unit Circle
******Unit Circle Plates******
Using the Unit Circle
sin θ =
csc θ =
cos θ =
sec θ =
tan θ =
cot θ =
Ex 1: Evaluate the following using the unit circle
a. cos 330°
b. sec 420°
c. sin 90°
d. tan −45°
e. tan 90°
Ex 2: Solve the trigonometric equations using the unit circle
1
a. cos 𝑥 = 2
b. sec 𝑥 = 2
Changes in Sign of Trig Functions
15
Ex 3: Given csc 𝜃 = − 7 , where 𝜃 is in Quadrant III. Find the other 5 trig ratios.
Ex 4: The terminal side of 𝜃 passes through (−4, 2). Find csc 𝜃 and tan 𝜃
5.6 The Law of Sines
Deriving the Law of Sines
Law of Sines can be used to solve _________ types of triangles (meaning ___________ or ____________)
Cases:
1.
2.
a.
b.
c.
Ex 1: Solve triangle ABC given A = 33°, B = 105° and b = 37.9. Round to the nearest hundredth.
Finding Area
Ex 2: Find the area to the nearest hundredth if A = 6°, B = 72° and b = 7 in.
5.8 The Law of Cosines
Only use when solving __________________ triangles
Cases:
1.
2.
The Law of Cosines
Ex 1: Find the missing angle measures and side lengths for triangle ABC. Round to the nearest hundredth.
a. C = 65°, a = 7cm, b = 5cm
b. a = 8in, b = 6in, c = 11in
5.7 The Ambiguous Case for the Law of Sines
Case 1: The given angle is acute
Case 2: The given angle is obtuse
Ex 1: Solve triangle ABC given B = 40°, b = 42m, c = 60m
Ex 2: Solve triangle ABC given C = 100°, a = 12cm, c = 21cm