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Math  /  Trigonometry

QuestionFind cos⁡1.59\cos 1.59 as +cos⁡x+\cos x or −cos⁡x-\cos x for 0<x<π20<x<\frac{\pi}{2} cos⁡1.59=\cos 1.59= □\square cos □\square

Studdy Solution

STEP 1

1. We are given an angle 1.591.59 radians and need to express cos⁡1.59\cos 1.59 in terms of cos⁡x\cos x where 0<x<π20 < x < \frac{\pi}{2}.
2. The angle 1.591.59 is in radians.
3. We will use trigonometric identities and properties of cosine to find the equivalent expression.

STEP 2

1. Determine the quadrant of the angle 1.591.59 radians.
2. Use the properties of cosine in different quadrants to express cos⁡1.59\cos 1.59 in terms of cos⁡x\cos x.

STEP 3

First, determine the quadrant in which the angle 1.591.59 radians lies. Recall that: - 00 to π2\frac{\pi}{2} radians is the first quadrant. - π2\frac{\pi}{2} to π\pi radians is the second quadrant. - π\pi to 3π2\frac{3\pi}{2} radians is the third quadrant. - 3π2\frac{3\pi}{2} to 2π2\pi radians is the fourth quadrant.
Since 1.591.59 radians is less than π\pi (approximately 3.143.14) and greater than π2\frac{\pi}{2} (approximately 1.571.57), it lies in the second quadrant.

STEP 4

In the second quadrant, the cosine of an angle is negative. Therefore, we can express cos⁡1.59\cos 1.59 as:
cos⁡1.59=−cos⁡(π−1.59)\cos 1.59 = -\cos(\pi - 1.59)
Now, calculate π−1.59\pi - 1.59:
π≈3.14159\pi \approx 3.14159 π−1.59≈3.14159−1.59=1.55159\pi - 1.59 \approx 3.14159 - 1.59 = 1.55159
Thus, we have:
cos⁡1.59=−cos⁡(1.55159)\cos 1.59 = -\cos(1.55159)
The expression for cos⁡1.59\cos 1.59 is:
cos⁡1.59=−cos⁡(1.55159)\cos 1.59 = -\cos(1.55159)

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