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Consider the relationship given below, for [tex]\frac{\pi}{2}\ \textless \ \theta\ \textless \ \pi[/tex]:

[tex]\sin^2 \theta + \cos^2 \theta = 1[/tex]

Which of the following best explains how this relationship and the value of [tex]\sin \theta[/tex] can be used to find the other trigonometric values?

A. The values of [tex]\sin \theta[/tex] and [tex]\cos \theta[/tex] represent the legs of a right triangle with a hypotenuse of 1; therefore, solving for [tex]\cos \theta[/tex] finds the unknown leg, and then all other trigonometric values can be found.

B. The values of [tex]\sin \theta[/tex] and [tex]\cos \theta[/tex] represent the angles of a right triangle; therefore, solving the relationship will find three angles of the triangle, and then all trigonometric values can be found.

C. The values of [tex]\sin \theta[/tex] and [tex]\cos \theta[/tex] represent the angles of a right triangle; therefore, other pairs of trigonometric ratios will have the same sum, 1, which can then be used to find all other values.

D. The values of [tex]\sin \theta[/tex] and [tex]\cos \theta[/tex] represent the legs of a right triangle with a hypotenuse of -1, since [tex]\theta[/tex] is in Quadrant II; therefore, solving for [tex]\cos \theta[/tex] finds the unknown leg, and then all other trigonometric values can be found.



Answer :

GinnyAnswer
Given the trigonometric identity
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
for an angle [tex]\(\theta\)[/tex] in the range [tex]\(\frac{\pi}{2} < \theta < \pi\)[/tex] (which means [tex]\(\theta\)[/tex] is in the second quadrant), let's examine the relationship and how it can be used to find other trigonometric values.

1. From the given trigonometric identity, if you know [tex]\(\sin \theta\)[/tex], you can solve for [tex]\(\cos \theta\)[/tex]:
[tex]\[ \cos^2 \theta = 1 - \sin^2 \theta \][/tex]
Since [tex]\(\theta\)[/tex] is in the second quadrant, [tex]\(\cos \theta\)[/tex] will be negative. Hence,
[tex]\[ \cos \theta = -\sqrt{1 - \sin^2 \theta} \][/tex]

2. Once we have [tex]\(\cos \theta\)[/tex], we can find the other trigonometric values:
- Tangent ([tex]\(\tan \theta\)[/tex]):
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \][/tex]
- Cosecant ([tex]\(\csc \theta\)[/tex]):
[tex]\[ \csc \theta = \frac{1}{\sin \theta} \][/tex]
- Secant ([tex]\(\sec \theta\)[/tex]):
[tex]\[ \sec \theta = \frac{1}{\cos \theta} \][/tex]
- Cotangent ([tex]\(\cot \theta\)[/tex]):
[tex]\[ \cot \theta = \frac{1}{\tan \theta} \][/tex]

Based on these steps, the best explanation that matches this process is:

The values of [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex] represent the legs of a right triangle with a hypotenuse of 1; therefore, solving for [tex]\(\cos \theta\)[/tex] finds the unknown leg, and then all other trigonometric values can be found.

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