# How do you use fundamental identities to find the values of the trigonometric values given tan(t) = -2.87 and csc(t) < 0?

Nov 19, 2016

$\sin t = - 0.95$, $\cos t = 0.33$, $\tan t = - 2.87$

$\cot t = - 0.35$, $\sec t = 3.04$ and $\csc t = - 1.05$

#### Explanation:

As $\tan \left(t\right) = - 2.87$ and $\csc \left(t\right) < 0$ means both are negative they lie in fourth quadrant. Hence, while $\sin t$ and $\cot t$ will be negative and $\cos t$ and $\sec t$ will be positive. Using these, we calculate all trigonometric ratios (rounding up to two places of decimals).

As $\tan t = - 2.87$, $\cot t = \frac{1}{-} 2.87 = - 0.35$ and

${\sec}^{2} t = 1 + {2.87}^{2} = 1 + 8.2369 = 9.2369$

hence, $\sec t = \sqrt{9.2369} = 3.04$ and $\cos t = 0.33$

and $\sin t = - \tan t \times \cos t = - 2.87 \times 0.33 = - 0.95$

and $\csc t = \frac{1}{-} 0.95 = - 1.05$

Hence, $\sin t = - 0.95$, $\cos t = 0.33$, $\tan t = - 2.87$

$\cot t = - 0.35$, $\sec t = 3.04$ and $\csc t = - 1.05$