Consider the relationship below, given $$ \frac{\pi}{2}

Question

Consider the relationship below, given $$ \frac{\pi}{2}<	heta<\pi $$ $$ \sin ^{2} 	heta+\cos ^{2} 	heta=1 $$ Which of the following best explains how this relationship and the value of $$ \sin 	heta $$ can be used to find the other trigonometric values?

UpStudy AI · Accepted Answer

1. We start with the Pythagorean identity:  
   $$
   \sin^2 \theta + \cos^2 \theta = 1
   $$
   
2. If the value of $$\sin \theta$$ is known and $$\frac{\pi}{2} < \theta < \pi$$ (which is quadrant II), we can solve for $$\cos \theta$$. Rearranging the identity gives:  
   $$
   \cos^2 \theta = 1 - \sin^2 \theta
   $$
   
3. Taking the square root of both sides, we have:  
   $$
   \cos \theta = \pm \sqrt{1 - \sin^2 \theta}
   $$
   However, since $$\theta$$ is in quadrant II where cosine is negative, we choose:  
   $$
   \cos \theta = -\sqrt{1 - \sin^2 \theta}
   $$
   
4. With $$\sin \theta$$ and $$\cos \theta$$ now computed, the remaining trigonometric functions can be found using their definitions. For example:  
   - Tangent is given by:  
     $$
     \tan \theta = \frac{\sin \theta}{\cos \theta}
     $$
   - Secant is the reciprocal of cosine:  
     $$
     \sec \theta = \frac{1}{\cos \theta}
     $$
   - Cosecant is the reciprocal of sine:  
     $$
     \csc \theta = \frac{1}{\sin \theta}
     $$
   - Cotangent is the reciprocal of tangent:  
     $$
     \cot \theta = \frac{\cos \theta}{\sin \theta}
     $$
     
5. Thus, knowing the value of $$\sin \theta$$ along with the Pythagorean identity allows us not only to compute $$\cos \theta$$ (taking into account the proper sign for the quadrant) but also to determine all other trigonometric values using standard definitions.
To find the other trigonometric values, use the Pythagorean identity $$ \sin^2 \theta + \cos^2 \theta = 1 $$. Knowing $$ \sin \theta $$ and the quadrant (II in this case), calculate $$ \cos \theta $$ as $$ -\sqrt{1 - \sin^2 \theta} $$. Then, use the definitions of tangent, secant, cosecant, and cotangent to find the remaining trigonometric values.

Consider the relationship below, given

Which of the following best explains how this relationship and the value of can be used to find the other
trigonometric values?

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Consider the relationship below, given Which of the following best explains how this relationship and the value of can be used to find the other trigonometric values?