77.7k views
1 vote
Could someone please help me solve this problem, please show evidence of the answer, process

Could someone please help me solve this problem, please show evidence of the answer-example-1
User Teekkari
by
7.8k points

1 Answer

6 votes

Answer:


\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)

User OTTA
by
8.1k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories