Using trigonometric identities, it is found that the sine and the tangent of the angle are given as follows:
- [tex]\sin{\theta} = \frac{1}{2}[/tex].
- [tex]\tan{\theta} = -\frac{\sqrt{3}}{3}[/tex]
How do we find the sine of an angle given the cosine?
We use the following identity:
[tex]\sin^{2}{\theta} + \cos^{2}{\theta} = 1[/tex]
In this problem, the cosine is:
[tex]\cos{\theta} = -\frac{\sqrt{3}}{2}[/tex]
Hence the sine is found as follows:
[tex]\sin^{2}{\theta} + \left(-\frac{\sqrt{3}}{2}\right)^2 = 1[/tex]
[tex]\sin^{2}{\theta} + \frac{3}{4} = 1[/tex]
[tex]\sin^{2}{\theta} = \frac{1}{4}[/tex]
[tex]\sin{\theta} = \pm \sqrt{\frac{1}{4}}[/tex]
Second quadrant, so the sine is positive, hence:
[tex]\sin{\theta} = \frac{1}{2}[/tex]
What is the tangent of an angle?
The tangent is given by the sine divided by the cosine, hence:
[tex]\tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}}[/tex]
Hence:
[tex]\tan{\theta} = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}}[/tex]
[tex]\tan{\theta} = -\frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}[/tex]
[tex]\tan{\theta} = -\frac{\sqrt{3}}{3}[/tex]
More can be learned about trigonometric identities at https://brainly.com/question/7331447
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