ALGEBRA 2 CHAPTER 13 NOTES Section 13-1 Right Triangle Trig

Objectives:

Understand and use trigonometric relationships of acute angles in triangles. CC.9-12.F.TF.3

Determine side lengths of right triangles by using trigonometric functions. CC.9-12.F.TF.5

A _____________________________ function is a function whose rule is

given by a trigonometric ratio.

A _______________________________________compares the lengths of

two sides of a right triangle.

The Greek letter theta ____ is traditionally used to represent the measure of an

acute angle in a right triangle.

SOH-CAH-TOA

Find the values of the six trigonometric functions for θ.

Special Right Triangles:

30-60-90:

70

24

45-45-90:

Use a trigonometric function to find the value of x.

A skateboard ramp will have a height of 12 in., and the angle between the ramp

and the ground will be 17°. To the nearest inch, what will be the length l of the ramp?

When an object is above or below another object, you can find distances indirectly by

using the _______________________ or the _________________________ between the objects.

ALGEBRA 2 CHAPTER 13 NOTES Section 13-2 Angles

Objectives: Draw angles in standard position.

Determine the values of the trigonometric functions for an angle in standard position.

An angle is in ___________________________when its vertex is at the origin and one ray is on the positive x-axis.

The _________________________of the angle is the ray on the x-axis.

The other ray is called the ___________________________ of the angle.

Positive Angles: Negative Angles:

Draw an angle with the given measure in standard position.

𝟑𝟐𝟎° − 𝟏𝟏𝟎° 𝟗𝟗𝟎°

____________________________________ are angles in standard position

with the same terminal side.

Find the measures of a positive angle and a negative angle that are coterminal with each given angle.

𝜽 = 𝟔𝟓° 𝜽 = 𝟒𝟏𝟎 °

For an angle θ in standard position, the _________________________ is the

positive acute angle formed by the terminal side of θ and the x-axis.

Find the measure of the reference angle for each given angle.

𝜽 = 𝟏𝟑𝟓° 𝜽 = −𝟏𝟎𝟓° 𝜽 = 𝟑𝟐𝟓°

Putting an angle on the xy-axis.

P(–3, 6) is a point on the terminal side of θ in standard position. Find the exact

value of the six trigonometric functions for θ.

ALGEBRA 2 CHAPTER 13 NOTES Section 13-3 Radian Measure and Unit Circle

Objectives: Convert angle measures between degrees and radians.

Find the values of trigonometric functions on the unit circle.

So far, you have measured angles in degrees. You can also measure angles in _______________________.

A _____________________ is a unit of angle measure based on arc length. Recall from geometry that an ___________ is an unbroken part of a circle. If a central angle

θ in a circle of radius r, then the measure of θ is defined as 1 radian

The circumference of a circle of radius r is _________.

Converting from degrees to radians:

Converting from radians to degrees:

A ____________________________________ is a circle with a radius of 1 unit. For every point P(x, y) on the unit circle, the value of r is 1. Therefore, for

an angle θ in the standard position:

How to determine the values of a trig function on a Unit Circle:

1. Draw the Angle on the xy plane

2. Determine the reference angle.

3. Construct a triangle with the x-axis and label the sides (30-60-90 or 45-45-

90).

4. Calculate using the appropriate trig ratio.

Another way to express the trigonometric functions:

Use a reference angle to find the exact value of the sine, cosine, and tangent of each angle

−𝟑𝟎°

ALGEBRA 2 CHAPTER 13 NOTES Section 13-3 Quadrantal Angles

Objectives: Construct basic sine, cosine and tangent graphs Calculate the values of the Quadrantal Angles

𝒚 = 𝒔𝒊𝒏 𝜽 𝒚 = 𝒄𝒐𝒔 𝜽 𝒚 = 𝒕𝒂𝒏 𝜽

𝒔𝒊𝒏 𝟐𝟕𝟎° 𝒄𝒐𝒔 − 𝟗𝟎° 𝒕𝒂𝒏 𝝅

𝟐 𝒄𝒔𝒄 𝝅

ALGEBRA 2 CHAPTER 13 NOTES Section 13-4 Inverse Trig Functions

Objectives: Evaluate inverse trigonometric functions.

Use trigonometric equations and inverse trigonometric functions to solve problems.

You have evaluated trigonometric functions for a given angle. You can also find the measure of angles given the value of a trigonometric function by using an

_____________________________________________ relation.

Find all of the possible values of the function: 𝒄𝒐𝒔−𝟏 √𝟑

𝟐

Find all of the possible values of the function: 𝒕𝒂𝒏−𝟏(−𝟏)

Principal Values of Inverse Trig Functions

𝒚 = 𝑺𝒊𝒏−𝟏𝒙 𝒚 = 𝑪𝒐𝒔−𝟏𝒙 𝒚 = 𝑻𝒂𝒏−𝟏𝒙 Evaluate each inverse trigonometric function. Give your answer in both radians and degrees

𝑪𝒐𝒔−𝟏 (−√𝟑

𝟐) 𝑨𝒓𝒄𝒕𝒂𝒏(−𝟏) 𝑺𝒊𝒏−𝟏 (−

√𝟐

𝟐)

𝑪𝒐𝒔−𝟏(𝟎)

ALGEBRA 2 CHAPTER 13 NOTES Section 13-5 Law of Sines

Objectives: Determine the area of a triangle given side-angle-side information.

Use the Law of Sines to find the side lengths and angle measures of a triangle.

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

Solve the triangle. Round all values to the nearest tenth.

Q

r

ALGEBRA 2 CHAPTER 13 NOTES Section 13-6 Law of Cosines

Objectives: Use the Law of Cosines to find the side lengths and angle measures of a triangle.

Use Heron’s Formula to find the area of a triangle.

Law of Cosines: Use the given measurements to solve ∆ABC. Round to the nearest tenth.

a = 8, b = 5, mC = 32.2°

a = 35, b = 42, c = 50.3