Question

Find cos248^o - sin212^o , if sin18^o = (√5 - 1)/4 

Answer 1

Let's solve the problem step by step:

1. Express cos 248° and sin 212° in terms of known angles:

* cos 248° = cos (180° + 68°) = -cos 68°
* sin 212° = sin (180° + 32°) = -sin 32°

So the expression becomes:
cos 248° - sin 212° = -cos 68° - (-sin 32°) = -cos 68° + sin 32°

2. Express cos 68° and sin 32° in terms of sin 18°:

* cos 68° = cos (90° - 22°) = sin 22°
* sin 32° = sin (18° + 14°)

We will use the following trigonometric identities:
* sin (a + b) = sin a cos b + cos a sin b
* sin (2a) = 2 sin a cos a
* cos (2a) = 1 - 2 sin^2 a
* cos (a) = sqrt(1-sin^2(a))

3. Express sin22 in terms of sin18
Sin(22) = sin(18+4)
Sin(22) is difficult to express directly in terms of sin18. To solve this problem a different approach is needed.

4. Alternative Approach:

We can rewrite the expression as:
cos 248° - sin 212° = -cos 68° + sin 32°

We can rewrite the expression as:

-cos(90-22) + sin(18+14) = -sin(22) + sin(18+14)

We can also write:
-cos(270-22) - sin(180+32) = -sin(22)+sin(32)

Lets use sin(32) = sin(2*16) and sin(22)= sin(2*11)

cos(248) - sin(212) = -cos(68) + sin(32)

We can express cos(68) as cos(90-22) = sin(22)
Then we have, -sin(22) + sin(32)
sin(32) - sin(22)

Use sin(A) - sin(B) = 2 cos((A+B)/2)sin((A-B)/2)

sin(32) - sin(22) = 2 cos(27) sin(5)

5. Express cos 27 and sin 5 in terms of sin18
This is very difficult to do. We are going to use a calculator to find the numeric values.

cos 248° ≈ -0.3746
sin 212° ≈ -0.5299

cos 248° - sin 212° ≈ -0.3746 - (-0.5299) ≈ 0.1553

Using a calculator:

sin(18) = (sqrt(5)-1)/4 ≈ 0.3090.

2cos(27)sin(5) ≈ 0.1553

Therefore, the value is approximately 0.1553.

Final Answer: The final answer is approximately 0.1553.