Question

For what angles c in [0, 2pi) does the cos(x) have the same value as sin (3pi/4)?

Answer 1

pi/4 and 7pi/4.

Explanation:

Sin 3pi/4 is in the second quadrant and is positive and has the same value as sin pi/4.

Sin (3pi/4) = sin (pi/4) = cos pi/4.

Also as cos x is positive in the 4th quadrant cos (2pi - pi/4)

= cos 7pi/4 is also equal to sin 3pi/4.

Answer 2

`(\pi)/(4)\,,\,(7\pi)/(4)`

Explanation:

Angle
`(3\pi)/(4)` lies in second quadrant in which
`\sin` is positive .

`\sin \left ( (3\pi)/(4) \right )\\=\sin \left ( \pi-(\pi)/(4) \right )\\=\sin \left ( (\pi)/(4) \right )\\=(1)/(√(2))`

We know that
`\cos` is positive in first and fourth quadrant .

In first quadrant :

We know that angle
`(\pi)/(4)` lies in first quadrant .

`\cos \left ( (\pi)/(4) \right )=(1)/(√(2))`

In fourth quadrant :

We know that angle
`(3\pi)/(4)` lies in fourth quadrant.

`\cos \left ( (7\pi)/(4) \right )\\=\cos \left ( 2\pi-(\pi)/(4) \right )\\=\cos \left ( (\pi)/(4) \right )\\=(1)/(√(2))`

So, for angles
`(\pi)/(4)\,,\,(7\pi)/(4)` ,
`\cos x` has the same value as
`\sin \left ( (3\pi)/(4) \right )`