Key to "Use an Addition or Subtraction Formula to write..."

SchachtN

Answered question

2021-09-13

Use an Addition or Subtraction Formula to write the expression as a trigonometric function of one number, and then find its exact value.

(\cos) \frac{13 π}{15} \cos (- \frac{π}{5}) - (\sin) \frac{13 π}{15} \sin (- \frac{π}{5})

Bertha Stark

Skilled2021-09-14Added 96 answers

Our expression is

(\cos) \frac{13 π}{15} \cos (- \frac{π}{5}) - (\sin) \frac{13 π}{15} \sin (- \frac{π}{5})

This kind of expression appears on the addition of angles formula for cosine

\cos (x + y) = \cos x \cos y - \sin x \sin y

with

x = \frac{13 π}{15}, y = - \frac{π}{5}

So we have

(\cos) \frac{13 π}{15} \cos (- \frac{π}{5}) - (\sin) \frac{13 π}{15} \sin (- \frac{π}{5}) = \cos (\frac{13 π}{15} + (- \frac{π}{5}))

= \cos (\frac{13 π}{15} - \frac{3 π}{15})

= (\cos) \frac{10 π}{15}

= (\cos) \frac{2 π}{3}

= - \cos (\frac{π}{3}) [\cos (π - x) = - \cos x]

= - \frac{1}{2}

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