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Mathematics
22.08.2021 22:47
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The angle θ 1 is located in Quadrant IV, and cos ⁡ ( θ 1 ) = 9/ 19 , theta, start

The angle θ 1 is located in Quadrant IV, and cos ⁡ ( θ 1 ) = 9/ 19 , theta, start subscript, 1, end subscript, right parenthesis, equals, start fraction, 9, divided by, 19, end fraction . What is the value of sin ⁡ ( θ 1 ) sin(θ 1 )sine, left parenthesis, theta, start subscript, 1, end subscript, right parenthesis? Express your answer exactly.
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jasmineharris121
jasmineharris121
4,5(97 marks)

sin\theta_1 = - \frac{2\sqrt{70}}{19}

Step-by-step explanation:

We are given that \theta_1 is in fourth quadrant.

cos\theta_1 is always positive in 4th quadrant and  

sin\theta_1 is always negative in 4th quadrant.

Also, we know the following identity about sin\theta and cos\theta:

sin^2\theta + cos^2\theta = 1

Using \theta_1 in place of \theta:

sin^2\theta_1 + cos^2\theta_1 = 1

We are given that cos\theta_1 = \frac{9}{19}

\Rightarrow sin^2\theta_1 + \dfrac{9^2}{19^2} = 1\\\Rightarrow sin^2\theta_1 = 1 - \dfrac{81}{361}\\\Rightarrow sin^2\theta_1 = \dfrac{361-81}{361}\\\Rightarrow sin^2\theta_1 = \dfrac{280}{361}\\\Rightarrow sin\theta_1 = \sqrt{\dfrac{280}{361}}\\\Rightarrow sin\theta_1 = +\dfrac{2\sqrt{70}}{19}, -\dfrac{2\sqrt{70}}{19}

\theta_1 is in 4th quadrant so sin\theta_1 is negative.

So, value of sin\theta_1 = - \frac{2\sqrt{70}}{19}

solismrY24
solismrY24
4,4(11 marks)
The second option is the answer.

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