Find the exact value of cos 8 degrees cos 38 degrees + sin 8 degrees sin 38 degrees

To find the exact value of cos 8 degrees cos 38 degrees + sin 8 degrees sin 38 degrees, we can use the trigonometric identity:

cos(A - B) = cos A cos B + sin A sin B

In this case, A = 8 degrees and B = 38 degrees. Plugging in these values, the equation becomes:

cos(8 - 38) = cos 8 cos 38 + sin 8 sin 38

cos(-30) = cos 8 cos 38 + sin 8 sin 38

We know that the cosine function is an even function, which means that cos(-x) = cos(x), so we can rewrite the equation as:

cos(30) = cos 8 cos 38 + sin 8 sin 38

Now we evaluate the values of cos 30 degrees, cos 8 degrees, and sin 8 degrees using the unit circle or a calculator:

cos 30 degrees = √3 / 2

cos 8 degrees ≈ 0.990

sin 8 degrees ≈ 0.141

Substituting the values into the equation, we get:

√3 / 2 = 0.990 * cos 38 + 0.141 * sin 38

To solve for cos 38 and sin 38, we can use another trigonometric identity:

sin² x + cos² x = 1

Rearranging the equation:

cos² 38 = 1 - sin² 38

Now, substitute sin 38 = √(1 - cos² 38) into the first equation:

√3 / 2 = 0.990 * cos 38 + 0.141 * √(1 - cos² 38)

Now we can square both sides of the equation to eliminate the square root:

(√3 / 2)² = (0.990 * cos 38 + 0.141 * √(1 - cos² 38))²

3 / 4 = (0.990 * cos 38 + 0.141 * √(1 - cos² 38))²

Expand the equation:

3 / 4 = 0.9801 * cos² 38 + 0.1962 * cos 38 √(1 - cos² 38) + 0.019881 * (1 - cos² 38)

Now, simplify the equation:

4 * (3 / 4) = 4 * (0.9801 * cos² 38 + 0.1962 * cos 38 √(1 - cos² 38) + 0.019881 * (1 - cos² 38))

3 = 3.9204 * cos² 38 + 0.7848 * cos 38 √(1 - cos² 38) + 0.079524 * (1 - cos² 38)

Now, let's solve for cos 38. Rearrange the equation:

0 = 3.9204 * cos² 38 + 0.7848 * cos 38 √(1 - cos² 38) + 0.079524 * (1 - cos² 38) - 3

This equation is a quadratic equation in terms of cos 38. Let's solve it using a calculator or numerical methods:

cos 38 ≈ 0.786
sin 38 ≈ 0.619

Now substituting the values of cos 38 and sin 38 back into the original equation:

cos 30 degrees = 0.990 * 0.786 + 0.141 * 0.619

cos 30 degrees ≈ 0.979

Therefore, the exact value of cos 8 degrees cos 38 degrees + sin 8 degrees sin 38 degrees is approximately 0.979.

To find the exact value of cos 8 degrees cos 38 degrees + sin 8 degrees sin 38 degrees, we can use the cosine of the sum formula.

The formula states that cos (x + y) = cos x cos y - sin x sin y.

In this equation, x is 8 degrees and y is 38 degrees. Therefore, we can rewrite the equation as:

cos (8 degrees + 38 degrees)

Now we can solve:

cos (8 degrees + 38 degrees) = cos 46 degrees

To find the cosine of 46 degrees, you can use a scientific calculator or table of trigonometric values. The exact value of cos 46 degrees is approximately 0.71934.

So, the exact value of cos 8 degrees cos 38 degrees + sin 8 degrees sin 38 degrees is approximately 0.71934.

Did you recognize the

cos (A-B) = cosAcosB + sinAsinB pattern ?

so cos8ºcos38º + sin8ºsin38º
= cos (8-38)
= cos (-30)
= cos 30
= √3/2