Question

In the following problem, the expression is the right side of the formula for cos (alpha - beta) with particular values for alpha and beta. cos (79 degree) cos (19 degree) + sin (79 degree) sin (19 degree)
Identify alpha and beta in each expression.
The value for alpha: degree
The value for beta: degree
Write the expression as the cosine of an angle. cos degree
Find the exact value of the expression. (Type an exact answer, using fraction, radicals and a rationalized denominator.)

Answer 1

Answer:

1.    $\alpha = 79$ and $\beta = 19$

2.   $cos(60)$

3.   $cos(60) = \frac{1}{2}$

Step-by-step explanation:

Given

$cos(\alpha - \beta )$

$cos(79)cos(19) + sin(79)sin(19)$

Solving for $\alpha$ and $\beta$

In trigonometry;

$cos(\alpha - \beta ) = cos\alpha\ cos\beta + sin\alpha\ sin\beta$

Equate the above expression to $cos(79)cos(19) + sin(79)sin(19)$

$cos(\alpha - \beta ) = cos\alpha\ cos\beta + sin\alpha\ sin\beta$ and $cos(\alpha - \beta ) = cos(79)cos(19) + sin(79)sin(19)$

By comparison

$cos\alpha\ cos\beta + sin\alpha\ sin\beta = cos(79)cos(19) + sin(79)sin(19)$

Compare expression on the right hand side to the left hand side

$cos\alpha\ cos\beta = cos(79)cos(19) \\\\ sin\alpha\ sin\beta = sin(79)sin(19)$

This implies that

$cos\alpha\ = cos(79)\\cos\beta = cos(19) \\\\ and\\\\sin\alpha\ = sin(79)\\sin\beta = sin(19)$

By further comparison

$\alpha = 79$ and $\beta = 19$

Substitute $\alpha = 79$ and $\beta = 19$ in $cos(\alpha - \beta )$

$cos(\alpha - \beta ) = cos(79 - 19)$

$cos(\alpha - \beta ) = cos(60)$

Hence, the expression is $cos(60)$

Solving for the exact values;

Express $cos(60)$ as a difference of angles

$cos(60) = cos(90 - 30)$

Recall that $cos(\alpha - \beta ) = cos\alpha\ cos\beta + sin\alpha\ sin\beta$

So;

$cos(90- 30 ) = cos(90) cos(30) + sin(90) sin(30)$

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In trigonometry;

$cos(90) = 0$; $cos(30) = \frac{\sqrt{3}}{{2}}$; $sin(90) = 1$; $sin(30) = \frac{1}{2}$;

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$cos(90- 30 ) = cos(90) cos(30) + sin(90) sin(30)$ becomes

$cos(90- 30 ) = 0 * \frac{\sqrt{3}}{2} + 1 * \frac{1}{2}$

$cos(90- 30 ) = 0 + \frac{1}{2}$

$cos(90- 30 ) = \frac{1}{2}$

Hence;

$cos(60) = \frac{1}{2}$