How to find the tangent for the missing side - ACT Math

Card 0 of 3

Question

For the right triangle shown below, what is the value of

$\frac{\sin\theta }{\cos\theta}$ ?

Answer

To solve this question, you must know SOHCAHTOA. This acronym can be broken into three parts to solve for the sine, cosine, and tangent.

$\textup{SOH: }\textup{Sine}=\frac{\textup{opposite}}{\textup{hypotenuse}}$

$\textup{CAH: }\textup{Cosine}=\frac{\textup{adjacent}}{\textup{hypotenuse}}$

$\textup{TOA: }\textup{Tangent}=\frac{\textup{opposite}}{\textup{adjacent}}$

We can use this information to solve our identity.

$\frac{\sin\theta }{\cos\theta}=\frac{\frac{\textup{opposite}}{\textup{hypotenuse}}}{\frac{adjacent}{hypotenuse}}$

Dividing by a fraction is the same as multiplying by its reciprocal.

$\frac{\frac{\textup{opposite}}{\textup{hypotenuse}}}{\frac{adjacent}{hypotenuse}}=\frac{\textup{opposite}}{\textup{hypotenuse}}\times \frac{\textup{hypotenuse}}{\textup{adjacent}}$

$\frac{\textup{opposite}}{\textup{hypotenuse}}\times \frac{\textup{hypotenuse}}{\textup{adjacent}}=\frac{\textup{oposite}}{\textup{adjacent}}$

The sine divided by cosine is the tangent of the angle.

$$Tan$ = \frac{\mbox{opposite}}{\mbox{adjacent}} = \frac{3}{4}$

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Question

What is the perimeter of the following figure?

Answer

The question asks for you to find the perimeter of the given figure. The figure has twelve sides total, of two varying lengths. One length is given to you, 4. The other length must be solved for using either the sine or tangent functions. However, one can arrive to answer more quickly by recognizing that the drawn triangle is actually a 3-4-5 triangle, where 3, 4, and 5 corresponds to each of the sides of the triangle. This is a pythagorean triple and this ratio should be easily remembered.

Thus if 3 is the missing side, and there are eight sides of length 3 and four sides of length 4, one can arrive to the answer:

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Question

A man is setting up a laser on the ground, angling it toward the very top of a flag pole. If the flag pole is high and the laser is placed away from its base, what should be the angle of the laser with the ground? (Answer in degrees, rounding to the nearest hundredth.)

Answer

You can draw out your scenario like a triangle:

Now, you know that this means:

$tan(x)=\frac{20}{12}$

Using your calculator, you can utilize the inverse function to calculate the degree measure of the angle:

$tan^-^1(\frac{20}{12})=59.03624346792652$

This rounds to $59.04^{\circ}$ degrees.

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