Question

Find the exact values of the remaining trigonometric functions of if terminates in Quadrant IV and tan() = −3/4.

Answer 1

We know that the angle terminates in the fourth quadrant and that the tangent of it is -3/4.

Angles that are on the fourth quadrant have a negative sine and a positive cosine.

We can start with the cot():

`\cot\theta=(1)/(\tan\theta)=(1)/(-(3)/(4))=-(4)/(3)`

We can relate the cosine with the tangent as:

`\begin{gathered} \tan\theta=(\sin\theta)/(\cos\theta)=(√(1-\cos^2\theta))/(\cos\theta) \\ -(3)/(4)=(√(1-\cos^2\theta))/(\cos\theta) \\ -(3)/(4)\cos\theta=√(1-\cos^2\theta) \\ (-(3)/(4)\cos\theta)^2=1-\cos^2\theta \\ (9)/(16)\cos^2\theta=1-\cos^2\theta \\ ((9)/(16)+1)\cos^2\theta=1 \\ (9+16)/(16)\cos^2\theta=1 \\ (25)/(16)\cos^2\theta=1 \\ \cos^2\theta=(16)/(25) \\ \cos\theta=\sqrt{(16)/(25)} \\ \cos\theta=(4)/(5) \end{gathered}`

We can now calculate the sine of the angle as:

`\begin{gathered} \sin\theta=\tan\theta\cdot\cos\theta \\ \sin\theta=-(3)/(4)\cdot(4)/(5)=-(3)/(5) \end{gathered}`

We can now calculate the sec() and csc() as:

`\begin{gathered} \sec\theta=(1)/(\cos\theta)=(5)/(4) \\ \\ \csc\theta=(1)/(\sin\theta)=-(5)/(3) \end{gathered}`

Answer:

sin() = -3/5

cos() = 4/5

cot() = -4/3

sec() = 5/4

csc() = -5/3