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Compound Angle Formulas: Derivation and Examples, Study notes of Algebra

Teesside University Algebra

This notebook contains the derivation of compound angle formulas for sine, cosine, and tangent using algebra and the unit circle. It also includes examples of how to use these formulas to find exact trigonometric values.

Typology: Study notes

2021/2022

Uploaded on 09/12/2022

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4.4CompoundAngleFormulas.notebook

May04,2014

4.4:CompoundAngleFormulas

•

Inthepagesthatfollow,compoundangleformulaswillbedeveloped

usingalgebraandtheunitcircle

•

Thecompoundangleformulasforsine,cosineandtangentare:

> sin(x+y)=sin(x)cos(y)+cos(x)sin(y)

> sin(x‐y)=sin(x)cos(y)‐cos(x)sin(y)

> cos(x+y)=cos(x)cos(y)‐sin(x)sin(y)

> cos(x‐y)=cos(x)cos(y)+sin(x)sin(y)

> tan(x+y)=tan(x)+tan(y)/1‐tan(x)tan(y)

> tan(x‐y)=tan(x)‐tan(y)/1+tan(x)tan(y)

•

Byusingcompoundangleformulas,youareabletodetermineexact

valuesfortrigonometricraosthatcanbeexpressedasthesumor

diﬀerenceofspecialangles.

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Download Compound Angle Formulas: Derivation and Examples and more Study notes Algebra in PDF only on Docsity!

4.4: Compound Angle Formulas

In the pages that follow, compound angle formulas will be developed using algebra and the unit circle
The compound angle formulas for sine, cosine and tangent are:

sin(x + y) = sin(x)cos(y) + cos(x)sin(y) sin(x ‐ y) = sin(x)cos(y) ‐ cos(x)sin(y) cos(x + y) = cos(x)cos(y) ‐ sin(x)sin(y) cos(x ‐ y) = cos(x)cos(y) + sin(x)sin(y) tan(x + y) = tan(x) + tan(y) / 1 ‐ tan(x)tan(y) tan(x ‐ y) = tan(x) ‐ tan(y) / 1 + tan(x)tan(y)
By using compound angle formulas, you are able to determine exact values for trigonometric raos that can be expressed as the sum or difference of special angles.

Proving the sum formula for cosine: cos(x+y)

Proving the sum formula for sin(x + y)

Recall: Co‐funcon identes involving (90 0 ‐ x)

Proving the difference formula for sin(x ‐ y) The difference formula for sine can be derived from the sum formula for sine:

Proving the difference formula for tan(x ‐ y)

Example 1: Find the exact value of

Attachments 4.4 example 1.pdf 4.4 example 2.pdf