Question

Could someone please help me solve this problem, please show evidence of the answer, process

Answer 1

`\tan \theta=-(√(5))/(2)`

`\cos \theta=(2)/(3)`

`\sin \theta=-(√(5))/(3)`

Explanation:

To find the exact values of the trigonometric functions of θ, we can use the following trigonometric identities:

`\boxed{\begin{minipage}{4 cm}\underline{Trigonometric Identities}\\\\$\sec x=(1)/(\cos x)$\\\\\\$\tan x=(\sin x)/(\cos x)$\\\\\\$\sin^2x+\cos^2x=1$\\\end{minipage}}`

In quadrant IV, cosine of the angle is positive, whereas sine of the angle is negative.

Given that sec θ = 3/2, and secant is the reciprocal of cosine:

`\sec \theta=(3)/(2)`

`(1)/(\cos \theta)=(3)/(2)`

`\boxed{\cos \theta=(2)/(3)}`

To find sin θ, we can substitute the found value of cos θ into the Pythagorean identity:

`\sin^2 \theta+\cos^2 \theta=1`

`\sin^2 \theta+\left((2)/(3)\right)^2=1`

`\sin^2 \theta=1-\left((2)/(3)\right)^2`

`\sin \theta=\sqrt{1-\left((2)/(3)\right)^2}`

`\sin \theta=(√(5))/(3)`

As sine of the angle is negative in quadrant IV:

`\boxed{\sin \theta=-(√(5))/(3)}`

Finally, substitute the values of sin θ and cos θ into the tan θ identity to find tan θ:

`\tan \theta=(\sin \theta)/(\cos \theta)`

`\tan \theta=(-(√(5))/(3))/((2)/(3))`

`\boxed{\tan \theta=-(√(5))/(2)}`

Therefore, the exact values of the trigonometric functions for θ are:

`\tan \theta=-(√(5))/(2)`

`\cos \theta=(2)/(3)`

`\sin \theta=-(√(5))/(3)`