Question

Suppose sin(theta) = -2/5 and cos(theta) is > 0. What is the value of tan(theta)?

Answer 1

The Solution:

The correct answer is -0.4364

Given that:

`\sin \theta=-(2)/(5)\text{ , and }\cos \theta>0`

We are required to find the value of:

`\tan \theta`

Step 1:

We shall find the angle represented with theta.

`\begin{gathered} \sin \theta=-(2)/(5) \\ \text{ Taking the }\sin ^(-1)\text{ of both sides, we get} \\ \\ \theta=\sin ^(-1)(-(2)/(5))=-23.5782^o \end{gathered}`

Recall:

The sine of angles is only negative in the 3rd and 4th quadrants.

And the formula for the 3rd quadrant is:

`\theta-180^o,180^o<\theta\leq270^o`

While the formula for the 4th quadrant is:

`360^o-\theta,270^o<\theta\leq360^o`

So,

`-\sin (\theta-180)=-\sin 23.5782^o`

This implies that:

`\theta-180=23.5782\text{ (having divided both sides by -}\sin )`

Collecting the like terms, we get

`\begin{gathered} \theta=23.5782+180=203.5782^o \\ \text{ }\theta\\e203.5782^o\text{ (since }\cos 203.5782^o\text{ is less than zero)} \\ \cos 203.5782=-0.9165<0 \\ \text{Therefore,} \\ \theta\\e203.5782^o \end{gathered}`

We shall use the 4th quadrant formula.

`-\sin (360^o-\theta)=-\sin 23.5782^o`

Dividing both sides by -sin, we get

`\begin{gathered} 360^o-\theta=23.5782 \\ \text{Collecting the like terms, we get} \\ 360-23.5782=\theta \\ \text{ So,} \\ \theta=336.4218^o \end{gathered}`

To check that it satisfies the given condition that says cos(theta) is greater zero, we have:

`\cos 336.4218^o=0.9165>0\text{ (Condition satisfied)}`

So, we have that:

`\theta=336.4218^o`

Step 2:

We shall find the value of tan(theta):

`\tan \theta=\tan (336.4218^o)=-0.4364`

Therefore, the correct answer is -0.4364