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Math  /  Trigonometry

QuestionSuppose that γ2\frac{\gamma}{2} is an angle in quadrant 1 and that cos⁡γ=161289\cos \gamma=\frac{161}{289}. Compute the exact value of sin⁡(γ2)\sin \left(\frac{\gamma}{2}\right). sin⁡(γ2)=23×06817\sin \left(\frac{\gamma}{2}\right)=23 \quad \times 0^{6} \frac{8}{17}

Studdy Solution

STEP 1

1. The angle γ2\frac{\gamma}{2} is in the first quadrant, where all trigonometric functions are positive.
2. We are given cos⁡γ=161289\cos \gamma = \frac{161}{289}.
3. We need to use the half-angle identity for sine to find sin⁡(γ2)\sin \left(\frac{\gamma}{2}\right).

STEP 2

1. Use the Pythagorean identity to find sin⁡γ\sin \gamma.
2. Use the half-angle identity for sine to find sin⁡(γ2)\sin \left(\frac{\gamma}{2}\right).

STEP 3

First, use the Pythagorean identity to find sin⁡γ\sin \gamma. The identity is:
sin⁡2γ+cos⁡2γ=1 \sin^2 \gamma + \cos^2 \gamma = 1
Substitute cos⁡γ=161289\cos \gamma = \frac{161}{289} into the identity:
sin⁡2γ+(161289)2=1 \sin^2 \gamma + \left(\frac{161}{289}\right)^2 = 1
Calculate (161289)2\left(\frac{161}{289}\right)^2:
(161289)2=16122892=2592183521 \left(\frac{161}{289}\right)^2 = \frac{161^2}{289^2} = \frac{25921}{83521}
Substitute back into the identity:
sin⁡2γ+2592183521=1 \sin^2 \gamma + \frac{25921}{83521} = 1
Solve for sin⁡2γ\sin^2 \gamma:
sin⁡2γ=1−2592183521 \sin^2 \gamma = 1 - \frac{25921}{83521} sin⁡2γ=83521−2592183521 \sin^2 \gamma = \frac{83521 - 25921}{83521} sin⁡2γ=5760083521 \sin^2 \gamma = \frac{57600}{83521}
Take the square root to find sin⁡γ\sin \gamma:
sin⁡γ=5760083521 \sin \gamma = \sqrt{\frac{57600}{83521}} sin⁡γ=240289 \sin \gamma = \frac{240}{289}

STEP 4

Use the half-angle identity for sine:
sin⁡(γ2)=1−cos⁡γ2 \sin \left(\frac{\gamma}{2}\right) = \sqrt{\frac{1 - \cos \gamma}{2}}
Substitute cos⁡γ=161289\cos \gamma = \frac{161}{289}:
sin⁡(γ2)=1−1612892 \sin \left(\frac{\gamma}{2}\right) = \sqrt{\frac{1 - \frac{161}{289}}{2}}
Calculate 1−1612891 - \frac{161}{289}:
1−161289=289−161289=128289 1 - \frac{161}{289} = \frac{289 - 161}{289} = \frac{128}{289}
Substitute back into the half-angle identity:
sin⁡(γ2)=1282892 \sin \left(\frac{\gamma}{2}\right) = \sqrt{\frac{\frac{128}{289}}{2}} sin⁡(γ2)=128578 \sin \left(\frac{\gamma}{2}\right) = \sqrt{\frac{128}{578}}
Simplify the fraction:
128578=64289 \frac{128}{578} = \frac{64}{289}
Take the square root:
sin⁡(γ2)=64289 \sin \left(\frac{\gamma}{2}\right) = \sqrt{\frac{64}{289}} sin⁡(γ2)=817 \sin \left(\frac{\gamma}{2}\right) = \frac{8}{17}
The exact value of sin⁡(γ2)\sin \left(\frac{\gamma}{2}\right) is:
817 \boxed{\frac{8}{17}}

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