cos 3pi/4 explai me testeps of exact formalr

Answer 1

To evaluate cos(3π/4), we can use the exact formula for cosines of angles in terms of the unit circle or trigonometric identities.

Step 1: Determine the reference angle
The reference angle is the acute angle formed between the terminal side of the angle and the x-axis. In this case, the reference angle is π/4 since the angle 3π/4 falls in the second quadrant and the terminal side is formed in the second quadrant as well.

Step 2: Determine the sign
In the second quadrant, cosine is negative. Therefore, cos(3π/4) is negative.

Step 3: Use the exact formula or trigonometric identity
The exact formula for cosine is cos(θ) = x, where θ is the angle and x is the x-coordinate on the unit circle.
In the second quadrant, the x-coordinate is negative and it can be found using the Pythagorean identity: sin^2(θ) + cos^2(θ) = 1 or cos(θ) = √(1 - sin^2(θ)).

Since the reference angle is π/4, sin(π/4) = √2/2. Therefore, cos(π/4) = √(1 - (√2/2)^2) = √(1 - 2/4) = √(1 - 1/2) = √(1/2) = 1/√2 = √2/2.

Finally, applying the sign from step 2, cos(3π/4) = -√2/2.

Therefore, cos(3π/4) is equal to -√2/2.

Answer 2

or, use the angle sum formula:
cos(3π/4) = cos(π - π/4) = cos(π)cos(π/4) + sin(π)sin(π/4) = (-1)(1/√2) = -1/√2

Answer 3

Apologies for the oversight. Yes, you can also use the angle sum formula to evaluate cos(3π/4).

Using the angle sum formula:
cos(3π/4) = cos(π - π/4) = cos(π)cos(π/4) + sin(π)sin(π/4)

Since cos(π) = -1 and sin(π) = 0, we can substitute these values:

cos(3π/4) = (-1)(cos(π/4)) + (0)(sin(π/4))
= -cos(π/4)
= -1/√2

Therefore, cos(3π/4) = -1/√2.