Question

If cos theta = k , 0 < k < 1 and theta is not a angle in the first quadrant then find the values of sin theta and tan theta in terms of k

Answer 1

Answer:

$sin(\theta)=\frac{-\sqrt{1-k^2}}{1},tan(\theta)=\frac{-\sqrt{1-k^2}}{k}$

Step-by-step explanation:

1) Since,

k is positive and hence cosine of theta.

=>$\theta$ is in fourth quadrant.

2)

$sin(\theta)$ and $tan(\theta)$  are negative.

We know that,

$sin^2(\theta)+cos^2(\theta)=1\\ \\=>sin^2(\theta)+k^2=1\\ \\=>sin^2(\theta)=1-k^2\\ \\=>sin(\theta)=-\sqrt{1-k^2}$

3) Also,

$1+tan^2(\theta)=sec^2(\theta)\\ \\=>tan^2(\theta)=sec^2(\theta)-1\\ \\=>tan^2(\theta)=\frac{1}{cos^2(\theta)}-1=\frac{1}{k^2}-1=\frac{1-k^2}{k^2}\\ \\ =>tan(\theta)=\frac{-\sqrt{1-k^2}}{k}$

Hence,

$\boxed{tan(\theta)=\frac{-\sqrt{1-k^2}}{k},sin(\theta)=-\sqrt{1-k^2}}$

Answer 2

Answer:

sin theta=-√1-k²

tan theta=-√1-k²/k