Video: Using the Addition Formula for Cosine Functions to Evaluate Trigonometric Expressions

Evaluate cos 55°45′ cos 79°15′ − sin 55°45′ sin 79°15′.

04:06

Video Transcript

Evaluate cos of 55 degrees and 45 minutes multiplied by cos of 79 degrees and 15 minutes minus sin 55 degrees 45 minutes multiplied by sin of 79 degrees 15 minutes.

Our first step here is to convert the minutes into degrees. We know that one degree is equal to 60 minutes. In this question, we have 45 minutes and 15 minutes that we need to convert into degrees. If we halve both sides of the conversion, we can see that 0.5 degrees is equal to 30 minutes. Halving this again tells us that 0.25 degrees is equal to 15 minutes. We can replace the 15 minutes in the expression with 0.25 degrees. 30 minutes plus 15 minutes is equal to 45 minutes. 0.5 degrees plus 0.25 degrees is equal to 0.75 degrees. Therefore, 45 minutes is equal to 0.75 degrees.

We can use these conversions to rewrite the expression in just degrees. Substituting in our values gives us cos of 55.75 multiplied by cos of 79.25 minus sin of 55.75 multiplied by sin of 79.25. All four of the angles are now given in degrees. One of the addition formulas for trigonometry states that cos of 𝛼 multiplied by cos of 𝛽 minus sin of 𝛼 multiplied by sin of 𝛽 is equal to cos of 𝛼 plus 𝛽. In our question, we can see that 𝛼 is equal to 55.75 degrees. 𝛽 is equal to 79.25 degrees. This means that our expression can be rewritten as cos of 55.75 degrees plus 79.25 degrees. Adding 55.75 and 79.25 gives us 135 degrees. The expression is simplified to cos of 135 degrees.

Let’s now consider our cosine or cos graph to help us calculate this value. The cosine graph is wave shaped and starts at one as the cos of zero is equal to one. As the graph is symmetrical, we can see that the cos of 135 degrees will be the negative value of the cos of 45 degrees. The cos of 45 degrees is one of our special angles. It is equal to one over root two.

This is also written as root two over two by rationalizing the denominator. As the cos of 135 degrees is the negative of this answer, it will be equal to negative one over root two or negative root two over two. The value of cos of 55 degrees and 45 minutes multiplied by cos of 79 degrees and 15 minutes minus the sin of 55 degrees 45 minutes multiplied by the sin of 79 degrees and 15 minutes is equal to negative one over root two or negative root two over two.

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