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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 (1.55159) □\square
Please answer all parts of the quesion.

Studdy Solution

STEP 1

1. We need to find an equivalent expression for cos⁡1.59\cos 1.59 in terms of cos⁡x\cos x where 0<x<π20 < x < \frac{\pi}{2}.
2. We need to determine whether cos⁡1.59\cos 1.59 is equal to +cos⁡x+\cos x or −cos⁡x-\cos x.
3. We need to evaluate cos⁡1.59\cos 1.59 and express it in terms of cos⁡(1.55159)\cos(1.55159).

STEP 2

1. Determine the quadrant in which the angle 1.591.59 radians lies.
2. Relate cos⁡1.59\cos 1.59 to cos⁡x\cos x for 0<x<π20 < x < \frac{\pi}{2}.
3. Evaluate cos⁡1.59\cos 1.59.
4. Express cos⁡1.59\cos 1.59 in terms of cos⁡(1.55159)\cos(1.55159).

STEP 3

Determine the quadrant for 1.591.59 radians:
The value 1.591.59 radians is less than π≈3.14159\pi \approx 3.14159 and greater than π2≈1.5708\frac{\pi}{2} \approx 1.5708.
Thus, 1.591.59 radians lies in the second quadrant.

STEP 4

In the second quadrant, the cosine of an angle is negative. Therefore, we have:
cos⁡1.59=−cos⁡x\cos 1.59 = -\cos x
where xx is the reference angle such that 0<x<π20 < x < \frac{\pi}{2}.

STEP 5

Evaluate cos⁡1.59\cos 1.59:
Using a calculator, we find:
cos⁡1.59≈−0.017452\cos 1.59 \approx -0.017452

STEP 6

Express cos⁡1.59\cos 1.59 in terms of cos⁡(1.55159)\cos(1.55159):
Since 1.591.59 is close to π2\frac{\pi}{2}, we can approximate:
1.59≈π2−1.551591.59 \approx \frac{\pi}{2} - 1.55159
Thus, cos⁡1.59\cos 1.59 can be expressed as:
cos⁡1.59=−cos⁡(1.55159)\cos 1.59 = -\cos(1.55159)
The value of cos⁡1.59\cos 1.59 is:
cos⁡1.59=−cos⁡(1.55159)\cos 1.59 = -\cos(1.55159)

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