Question

1. sin pi/8 - sin pi/4 2. cos pi/10 + sin pi/10

Answer 1

Відповідь:

1) sinπ/10 + sinπ/5 = 2sin3π/20 cosπ/20

2) cosπ/8 + cosπ/16 = 2cos3π/32 cosπ/32

Пояснення:

Answer 2

Let's calculate the given expressions step by step:

1. sin(pi/8) - sin(pi/4): To simplify this expression, we can use the trigonometric identity: sin(a) - sin(b) = 2 * cos((a + b)/2) * sin((a - b)/2).

Using this identity, we have: sin(pi/8) - sin(pi/4) = 2 * cos((pi/8 + pi/4)/2) * sin((pi/8 - pi/4)/2) = 2 * cos(3pi/8) * sin(-pi/8) = 2 * cos(3pi/8) * (-sin(pi/8))

Now, let's substitute the values of sin(pi/8) and cos(3pi/8) from trigonometric tables or calculators: sin(pi/8) ≈ 0.3827 cos(3pi/8) ≈ 0.9239

Therefore, the expression becomes: 2 * 0.9239 * (-0.3827) ≈ -0.7065

So, sin(pi/8) - sin(pi/4) ≈ -0.7065.

2. cos(pi/10) + sin(pi/10): Here, we can directly evaluate the cosine and sine values using trigonometric tables or calculators: cos(pi/10) ≈ 0.9848 sin(pi/10) ≈ 0.1736

Therefore, the expression becomes: 0.9848 + 0.1736 ≈ 1.1584

So, cos(pi/10) + sin(pi/10) ≈ 1.1584.

To summarize:

1. sin(pi/8) - sin(pi/4) ≈ -0.7065
2. cos(pi/10) + sin(pi/10) ≈ 1.1584