Question

suppose theta is an angle in the standard position whose terminal side is in Quadrant 1 and sin theta = S4/S5. find the exact values of the five remaining trig functions of theta

Answer 1

Answer:

$csc(\theta)=\frac{5}{4}$

$cos(\theta)=\frac{3}{5}$

$sec(\theta)=\frac{5}{3}$

$tan(\theta)=\frac{4}{3}$

$cot(\theta)=\frac{3}{4}$

Step-by-step explanation:

step 1

Find the $csc(\theta)$

we know that

$csc(\theta)=\frac{1}{sin(\theta)}$

we have

$sin(\theta)=\frac{4}{5}$

therefore

$csc(\theta)=\frac{5}{4}$

step 2

Find the $cos(\theta)$

we know that

$sin^2(\theta)+cos^2(\theta)=1$

we have

$sin(\theta)=\frac{4}{5}$

substitute

$(\frac{4}{5})^2+cos^2(\theta)=1$

$cos^2(\theta)=1-(\frac{4}{5})^2$

$cos^2(\theta)=1-(\frac{16}{25})$

$cos^2(\theta)=(\frac{9}{25})$

square root both sides

$cos(\theta)=\pm(\frac{3}{5})$

Remember that the angle theta is in quadrant I

so

The value of cosine of angle theta is positive

$cos(\theta)=\frac{3}{5}$

step 3

Find the $sec(\theta)$

we know that

$sec(\theta)=\frac{1}{cos(\theta)}$

we have

$cos(\theta)=\frac{3}{5}$

therefore

$sec(\theta)=\frac{5}{3}$

step 4

Find the value of  $tan(\theta)$

we know that

$tan(\theta)=\frac{sin(\theta)}{cos(\theta)}$

we have

$sin(\theta)=\frac{4}{5}$

$cos(\theta)=\frac{3}{5}$

substitute the values

$tan(\theta)=\frac{4}{3}$

step 5

Find the value of  $cot(\theta)$

we know that

$cot(\theta)=\frac{1}{tan(\theta)}$

we have

$tan(\theta)=\frac{4}{3}$

therefore

$cot(\theta)=\frac{3}{4}$