To find the values of f(5) and f(1) for the function f(x) = sin8(x) imes cos8(x), let’s break it down step-by-step.
Step 1: Understand the function
The function is a product of the sine and cosine functions raised to the eighth power. This means the value of the function will depend on the sine and cosine values at the given input.
Step 2: Calculate f(5)
First, we need to calculate sin(5) and cos(5). These values are typically computed using a calculator or by referring to trigonometric tables. Assuming we have these values:
- sin(5) ≈ -0.9589
- cos(5) ≈ 0.2837
Now, we can plug these values into the function:
f(5) = sin8(5) imes cos8(5)
Calculating this gives:
- sin8(5) ≈ (-0.9589)8 ≈ 0.8590
- cos8(5) ≈ (0.2837)8 ≈ 0.0006
Thus, we get:
f(5) ≈ 0.8590 imes 0.0006 ≈ 0.000514
Step 3: Calculate f(1)
Next, we proceed to find f(1). Again, we calculate sin(1) and cos(1):
- sin(1) ≈ 0.8415
- cos(1) ≈ 0.5403
Plugging these values into the function gives us:
f(1) = sin8(1) imes cos8(1)
Calculating this yields:
- sin8(1) ≈ (0.8415)8 ≈ 0.4192
- cos8(1) ≈ (0.5403)8 ≈ 0.0259
Thus, we conclude:
f(1) ≈ 0.4192 imes 0.0259 ≈ 0.0109
Conclusion:
After calculating both required values, we find:
- f(5) ≈ 0.000514
- f(1) ≈ 0.0109
This method should help you understand how to approach similar problems involving trigonometric functions raised to powers!