Sum Of Angle Formulas For Sine & Cosine

Hi there. In this math education post I cover the topic of sum of angle formulas for sine and cosine. This topic is a part of the trigonometry section of mathematics. You normally find this in high school mathematics (in the Toronto area at least).

It is assumed that the reader is familiar with trigonometric ratios and the CAST rule. A calculator can be used instead of the CAST rule.

These formulas are helpful for when calculators are not allowed nor available.

Math text rendered in LaTeX (lay-tech) with Quicklatex.com.

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Topics

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• Sum Of Angle Formulas For Sine
• Sum Of Angle Formulas For Cosine
• Examples

 

Sum Of Angle Formulas For Sine

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There is a formula for the sine of the sum of angles. The sum of angles is an input for the sine function. I use A and B as the different angle amounts. Other formulas use the greek letters alpha (α) and beta (β).

 

You can also have A minus B as a difference of angles. The formula for this case would be

 

Example One

What is the sine of 105 degrees?

It is a good practice to decompose 105 degrees into two common angles such as 30 degrees, 60 degrees, 45 degrees and its multiples. For this example 105 degrees can be split into 60 degrees and 45 degrees. Angle A would be 60 degrees and angle B would be 45 degrees.

 

Compute each of the sine and cosine values and perform order of operations. The sine of 60 is half of square root of three and the cosine of 60 degrees is one half. The cosine of 45 degrees is equal to the sine of 45 degrees which is one divided by square root 2.

 

Example Two

Compute the sine of 150 degrees.

Common angles are 30 degrees, 45 degrees and 60 degrees. None of these three really work so you have to use either multiples of 30, 45 or 60 degrees or other common angles like 90 degrees and 180 degrees.

For here I use 180 degrees minus 30 degrees to obtain 150 degrees. Use the difference of angles formula for sine this time.

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Sum Of Angle Formulas For Cosine

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The sum of angle formulas for cosine is kind of similar to the sine version. Here are the two formulas for the sum of angles and difference of angles.

 

Example One

What is the value of the cosine of 105 degrees?

Decompose 105 degrees into 60 degrees and 45 degrees. Angle A is 60 degrees and angle B is 45 degrees.
 

 

Example Two

Determine the cosine of negative fifteen degrees.

From the known common angles of 30 degrees, 45 degrees, 60 degrees and 90 degrees, how can negative fifteen degrees be expressed as a difference?

For negative 15 degrees I got 30 minus 45 degrees. Let angle A be 30 degrees and angle B be 45 degrees.

 

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Thank you for reading.

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