Question

consider the relationship below, given $\frac{pi}{2}<\theta$sin^{2}\theta+cos^{2}\theta = 1$
which of the following best explains how this relationship and the value of $sin\theta$ can be used to find the other trigonometric values?
the values of $sin\theta$ and $cos\theta$ represent the legs of a right triangle with a hypotenuse of 1; therefore, solving for $cos\theta$ finds the unknown leg, and then all other trigonometric values can be found.
the values of $sin\theta$ and $cos\theta$ represent the angles of a right triangle; therefore, solving the relationship will find all three angles of the triangle, and then all trigonometric values can be found.
the values of $sin\theta$ and $cos\theta$ 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.
the values of $sin\theta$ and $cos\theta$ represent the legs of a right triangle with a hypotenuse of - 1, since $\theta$ is in quadrant ii; therefore, solving for $cos\theta$ finds the unknown leg, and then all other trigonometric values can be found.

Explanation:

In a right - triangle context for trigonometry, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$ and $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Given $\sin^{2}\theta+\cos^{2}\theta = 1$ and the hypotenuse of the unit - circle right - triangle is 1, $\sin\theta$ and $\cos\theta$ are the lengths of the legs. Solving for $\cos\theta$ when $\sin\theta$ is known gives the other leg length, and then other trigonometric functions (such as $\tan\theta=\frac{\sin\theta}{\cos\theta}$, $\csc\theta=\frac{1}{\sin\theta}$, $\sec\theta=\frac{1}{\cos\theta}$, $\cot\theta=\frac{\cos\theta}{\sin\theta}$) can be found. The hypotenuse is always positive 1 in the unit - circle definition, and $\sin\theta$ and $\cos\theta$ are not angles but side - length ratios.

Answer:

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