Angles and the Trigonometric Functions

Trigonometric Functions graphs, Inverse Trigonometric Functions

Trigonometric Identities and Equations

Laws of Sines, Cosines and Vectors

Complex Numbers, Polar Coordinates and Parametric Equations

In Exercises 35–38, find the exact value of the following under the given conditions:

β
e. cos -------
2

1 3𝝅 1 3𝝅 
sin α =﹣ ------ , 𝝅 < α < ------- , and cos β =﹣------ , 𝝅 < β < ---------. 
3 2 3 2

Law of cosines - Trig identities and examples

Law of Cosines, Example 5

Oblique Word Problems

The Law of Cosines

Application of the Law of Cosines

Law of Cosines Word Problem

Heron's Formula - Example

Using the Law of Cosines (SSS) to find an angle

In Exercises 35–38, find the exact value of the following under the given conditions:

β
e. cos -------
2

3 𝝅 12 𝝅 
sin α = ------- , 0 < α < -------- , and sin β = --------- , --------- < β < 𝝅. 
5 2 13 2

Heron's Formula to Find Height of a Triangle

In Exercises 5–8, each expression is the right side of the formula for cos (α - β) with particular values for α and β. 

c. Find the exact value of the expression.

cos 50° cos 20° + sin 50° sin 20°

Using Law of Cosines

Law of Cosines, Word Problem #1

In Exercises 57–64, find the exact value of the following under the given conditions:

a. cos (α + β)

3 1
tan α = ﹣ ------ , α lies in quadrant II, and cos β = ------- , β lies in quadrant I.
4 3

In Exercises 57–64, find the exact value of the following under the given conditions:

a. cos (α + β)

3 3𝝅 1 3𝝅 
tan α = ------ , 𝝅 < α < -------- , and cos β = ------- , ---------- < β < 2𝝅.
4 2 4 2

How to Find the Area of a Triangle with Trigonometry

Law of Cosines, Example 1

Be sure that you've familiarized yourself with the first set of formulas presented in this section by working C1–C4 in the Concept and Vocabulary Check. In Exercises 1–8, use the appropriate formula to express each product as a sum or difference.

cos 7x cos 3x

In Exercises 5–8, each expression is the right side of the formula for cos (α - β) with particular values for α and β. 

b. Write the expression as the cosine of an angle.

5π π 5π π
cos ------- cos -------- + sin -------- sin -------
12 12 12 12

In Exercises 5–8, each expression is the right side of the formula for cos (α - β) with particular values for α and β. 

b. Write the expression as the cosine of an angle.

cos 50° cos 20° + sin 50° sin 20°

Heron's formula - Perimeter, area, and volume

In Exercises 5–8, each expression is the right side of the formula for cos (α - β) with particular values for α and β. 

c. Find the exact value of the expression.

5π π 5π π
cos ------- cos -------- + sin -------- sin -------
12 12 12 12

Use one or more of the six sum and difference identities to solve Exercises 13–54.

In Exercises 13–24, find the exact value of each expression.

cos(135° + 30°)

Heron's Formula, Example 3

In Exercises 35–38, find the exact value of the following under the given conditions:

b. cos(α﹣β)

1 3𝝅 1 3𝝅 
sin α =﹣ ------ , 𝝅 < α < ------- , and cos β =﹣------ , 𝝅 < β < ---------. 
3 2 3 2

Heron's Formula: Area of a Triangle

How to use heron's formula to find the area of a triangle with SSS

Heron's Formula, Example 1

Heron's Formula

Be sure that you've familiarized yourself with the first set of formulas presented in this section by working C1–C4 in the Concept and Vocabulary Check. In Exercises 1–8, use the appropriate formula to express each product as a sum or difference.

sin 6x sin 2x

Use the Law of Cosines to Solve Oblique Triangles

Law of Cosines, Example 3

In Exercises 57–64, find the exact value of the following under the given conditions:

a. cos (α + β)

8 1
cos α = ------ , α lies in quadrant IV, and sin β = ﹣------- , β lies in quadrant III.
17 2

Use the following conditions to solve Exercises 1–4:

4 𝝅
sin α = ----- , ------- < α < 𝝅
5 2

5 𝝅
cos β = ------ , 0 < β < ------
13 2

Find the exact value of each of the following.

cos (α + β)

Law of Cosines

The Cosine Rule (2 of 3: Basic Example of Finding Sides)

Using the law of cosines for a triangle with SAS

Use the formula for the cosine of the difference of two angles to solve Exercises 1–12. 

In Exercises 1–4, find the exact value of each expression.

cos(45° - 30°)

Find the Bearing and Distance - Law of Cosines word problem

In Exercises 57–64, find the exact value of the following under the given conditions:

a. cos (α + β)

3 5
sin α = ------ , α lies in quadrant I, and sin β = ------- , β lies in quadrant II.
5 13

Heron's Formula 2

In Exercises 35–38, find the exact value of the following under the given conditions:

b. cos(α﹣β)

3 𝝅 12 𝝅 
sin α = ------- , 0 < α < -------- , and sin β = --------- , --------- < β < 𝝅. 
5 2 13 2

Law of Cosines, Example 4

In Exercises 14–19, use a sum or difference formula to find the exact value of each expression.

cos(45° + 30°)

Law of Cosines, Example 2

Law of Cosines, Example 6

In Exercises 69–74, rewrite each expression as a simplified expression containing one term.

cos (α + β) cos β + sin (α + β) sin β

This course covers topics including trigonometric functions (with graphs and inverse functions), identities and equations, solutions of triangles, complex numbers and more.

College Trigonometry

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Use the Law of Sines to Solve Oblique Triangles

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Solve Applied Problems Using the Law of Sines

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