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ACT Math : How to find positive tangent

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Example Question #1 : How To Find Positive Tangent

Math2

For triangle $\DeltaABC$ $\displaystyle \DeltaABC$ $\Delta ABC$ $\displaystyle \Delta ABC$ , what is the cotangent of angle $\angle B$ $\displaystyle \angle B$ ?

Possible Answers:

$\frac{d}{e}$ $\displaystyle \frac{d}{e}$

$\frac{d}{f}$ $\displaystyle \frac{d}{f}$

$\frac{f}{d}$ $\displaystyle \frac{f}{d}$

$\frac{f}{e}$ $\displaystyle \frac{f}{e}$

Correct answer:

$\frac{f}{d}$ $\displaystyle \frac{f}{d}$

Explanation:

The cotangent of the angle of a triangle is the adjacent side over the opposite side. The answer is $\frac{f}{d}$ $\displaystyle \frac{f}{d}$

Math2-p1

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Example Question #2 : How To Find Positive Tangent

What is the tangent of the angle formed between the origin and the point $\displaystyle (-12,-5)$ if that angle is formed with one side of the angle beginning on the $\displaystyle x$ -axis and then rotating counter-clockwise to $\displaystyle (-12,-5)$ ? Round to the nearest hundredth.

Possible Answers:

-0.42 $\displaystyle -0.42$

-2.4 $\displaystyle -2.4$

0.38 $\displaystyle 0.38$

2.4 $\displaystyle 2.4$

0.42 $\displaystyle 0.42$

Correct answer:

0.42 $\displaystyle 0.42$

Explanation:

Recall that when you calculate a trigonometric function for an obtuse angle like this, you always use the $\displaystyle x$ -axis as your reference point for your angle. (Hence, it is called the "reference angle.")

Now, it is easiest to think of this like you are drawing a little triangle in the third quadrant of the Cartesian plane. It would look like:

Tan125

So, the tangent of an angle is:

$\frac{opposite}{adjacent}$ $\displaystyle \frac{opposite}{adjacent}$ or, for your data, $\frac{5}{12}$ $\displaystyle \frac{5}{12}$ .

This is $\displaystyle 0.41666666666667$ . Rounding, this is 0.42 $\displaystyle 0.42$ . Since $\displaystyle (-12,-5)$ is in the third quadrant, your value must be positive, as the tangent function is positive in that quadrant.

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Example Question #23 : Trigonometry

What is the tangent of the angle formed between the origin and the point $\displaystyle (-5,-16)$ if that angle is formed with one side of the angle beginning on the $\displaystyle x$ -axis and then rotating counter-clockwise to $\displaystyle (-5,-16)$ ? Round to the nearest hundredth.

Possible Answers:

5.14 $\displaystyle 5.14$

-3.2 $\displaystyle -3.2$

3.2 $\displaystyle 3.2$

0.31 $\displaystyle 0.31$

-0.31 $\displaystyle -0.31$

Correct answer:

3.2 $\displaystyle 3.2$

Explanation:

Now, it is easiest to think of this like you are drawing a little triangle in the third quadrant of the Cartesian plane. It would look like:

Tan516

So, the tangent of an angle is:

$\frac{opposite}{adjacent}$ $\displaystyle \frac{opposite}{adjacent}$ or, for your data, $\frac{16}{5}$ $\displaystyle \frac{16}{5}$ , or 3.2 $\displaystyle 3.2$ . Since $\displaystyle (-5,-16)$ is in the third quadrant, your value must be positive, as the tangent function is positive in this quadrant.

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Example Question #1 : How To Find Positive Tangent

A ramp is being built at an angle of $30^{\circ}$ $\displaystyle 30^{\circ}$ from the ground. It must cover $\displaystyle 10$ horizontal feet. What is the length of the ramp? Round to the nearest hundredth of a foot.

Possible Answers:

$13.53\:ft$ $\displaystyle 13.53\:ft$

$11.55\:ft$ $\displaystyle 11.55\:ft$

$16.91\:ft$ $\displaystyle 16.91\:ft$

$12.41\:ft$ $\displaystyle 12.41\:ft$

$5.77\:ft$ $\displaystyle 5.77\:ft$

Correct answer:

$11.55\:ft$ $\displaystyle 11.55\:ft$

Explanation:

Based on our information, we can draw this little triangle:

Tan10

Since we know that the tangent of an angle is $\frac{opposite}{adjacent}$ $\displaystyle \frac{opposite}{adjacent}$ , we can say:

$tan(30)=\frac{y}{10}$ $\displaystyle tan(30)=\frac{y}{10}$

This can be solved using your calculator:

$\displaystyle y=10tan(30)$ or $\displaystyle 5.77350269189626$

Now, to solve for $\displaystyle r$ , use the Pythagorean Theorem, $\displaystyle a^2+b^2=c^2$ , where $\displaystyle a$ and $\displaystyle b$ are the legs of a triangle and $\displaystyle c$ is the triangle's hypotenuse. Here, c=r $\displaystyle c=r$ , so we can substitute that in and rearrange the equation to solve for $\displaystyle r$ :

$r=\sqrt{a^2+b^2}$ $\displaystyle r=\sqrt{a^2+b^2}$

Substituting in the known values:

$r=\sqrt{10^2+5.77350269189626^2}$ $\displaystyle r=\sqrt{10^2+5.77350269189626^2}$ , or approximately $\displaystyle 11.54700538379252$ . Rounding, this is 11.55 $\displaystyle 11.55$ .

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ACT Math : How to find positive tangent

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : How To Find Positive Tangent

Example Question #2 : How To Find Positive Tangent

Example Question #23 : Trigonometry

Example Question #1 : How To Find Positive Tangent

All ACT Math Resources

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