More Lessons for Trigonometry

Math Worksheets

A series of free, online Trigonometry Video Lessons.

Videos, worksheets, and activities to help Trigonometry students.

In this lesson, we will learn

- the cosine addition formula
- how to derive the cosine of a sum and difference of two angles
- the sine addition formula
- how to derive the sine of a sum and difference of two angles
- how to use the sine and cosine addition and subtraction formulas to prove identities
- how to use the sine and cosine addition and subtraction formulas to determine function values.

This video shows the formula for deriving the cosine of a sum of two angles.

cos(A + B) = cosAcosB − sinAsinB

We will use the unit circle definitions for sine and cosine, the Pythagorean identity, the distance formula.

A proof that cos (A − B) = cosAcosB + sinAsinB.

The main idea is to create a triangle whose angle is a difference of two other angles, whose adjacent sides, out of simplicity, are both 1. By using both the distance formula and the law of cosines, we can get an equation where cos(A − B) is present.

How to derive the sine of a sum formula. We will use the cofunction identities and the cosine of a difference formula.

sin(A + B) = sinAcosB + cosAsinB
The derivation of the sum and difference identities for cosine and sine.

cos(A + B) = cosAcosB − sinAsinB

cos (A − B) = cosAcosB + sinAsinB

sin(A + B) = sinAcosB + cosAsinB

sin(A − B) = sinAcosB − cosAsinB

Using the formulas,

we see that sin(π/2-x) = cos(x), cos(π/2-x) = sin(x);

that sin(x + π) = −sin(x), cos(x + π) = −cos(x);

and that sin(π − x) = sin(x), cos(π − x) = −cos(x).

The formulas also give the tangent of a difference formula, for tan(alpha − beta).

How to use the sine and cosine addition formulas to prove the cofunction identities. In this video, we will verify a Trig identity by using the basic identities and the identity for the cosine of a sum or difference. We prove that (cos(x+y))/(cos(x − y)) = (cot y − tan x)/(cot y + tan x) using a trig proof.

Determine the exact value of sin(105 degrees) using the sum and difference identity.

Determine the exact value of cos(13/12π) using the sum and difference identity.

Use the sum and difference identities for sine to determine the function values.

Determine sin(A − B), given sin A = 4/5 in quadrant II and cos B = −5/13 in quadrant III.

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