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