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Trigonometry : Solving Triangles

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Example Question #1 : Finding Sides

In a right triangle, the legs are and . What is the length of the hypotenuse?

Possible Answers:

500

Correct answer:

Explanation:

Here, we can use the Pythagorean Theorem.

This states that the sum of the squares of the legs is equal to the square of the hypotenuse.

Thus, we plug into the formula to get our hypotenuse.

$7^{^{2}}+24^{2} = c^{2}$

$c=\sqrt{49+576}=\sqrt{625}=25$

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Example Question #2 : Finding Sides

If the altitude of a triangle has a height of 5, and also bisects a 60 degree angle, what must be the perimeter of the full triangle?

Possible Answers:

$\frac{ 10\sqrt3}{3}$

$\frac{5\sqrt3}{3}$

$5\sqrt3$

$10\sqrt3$

$2\sqrt3$

Correct answer:

$10\sqrt3$

Explanation:

The altitude of the triangle splits the 60 degree angle into two 30 degree angles. Notice that this will create two right triangles with 30-60-90 angles. This indicates that the full triangle is an equilateral triangle.

The altitude represents the adjacent side of the split triangles. The bisected angle is 30 degrees. Find the hypotenuse of either split triangle. This is the side length of the equilateral triangle.

$cos(\theta)=\frac{A}{H}$

$H=\frac{5}{cos(30)}= \frac{5}{\frac{\sqrt3}{2}}= \frac{5 \times 2}{\sqrt{3}} = \frac{10}{\sqrt3}$

Rationalize the denominator.

$\frac{10}{\sqrt3}\times\frac{\sqrt3}{\sqrt3}=\frac{ 10\sqrt3}{3}$

Since the full triangle has three sides, multiply this value by three for the perimeter.

$3\left(\frac{ 10\sqrt3}{3}\right)=10\sqrt3$

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Example Question #3 : Finding Sides

A horizontal ramp of unknown length meets the ground at angle of 20°. The height of the ramp is 6 feet and makes a right angle with the ground. What is the length of the ramp?

Possible Answers:

None of the other answers.

$15.5\:\uptext{ft}$

$20.5\:\uptext{ft}$

$25.5\:\uptext{ft}$

$22.4\:\uptext{ft}$

Correct answer:

None of the other answers.

Explanation:

The question is looking for the length of the ramp which would be the length of the part one would walk on and is also synonymous with the hypotenuse of our right triangle. With the given angle and opposite side we can use the sine function to solve for the length of the ramp because

$\sin\theta =\frac{opp}{hyp}$ .

We have all but the length of the hypotenuse.

Solve for the hypotenuse:

$hyp=\frac{opp}{\sin\theta }$

Plug in the angles and sides given:

$hyp=\frac{6}{\sin20 }=17.5\:\uptext{ft}$

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Example Question #4 : Finding Sides

A zipline is attached from the edge of the roof of a 23 foot high building and meets the ground 23 feet from the base of the building. Find the angle $\theta$ of elevation the cable makes with the ground.

Possible Answers:

$\theta=34^{\circ}$

$\theta=49^{\circ}$

$\theta=25^{\circ}$

$\theta=45^{\circ}$

$\theta=35^{\circ}$

Correct answer:

$\theta=45^{\circ}$

Explanation:

Concerning the angle of elevation which we are to find, we know the side length opposite this angle and the side length adjacent to it.

We use

$\tan=\frac{opp}{adj}$ .

$\tan\theta=\frac{opp}{adj}=\frac{23}{23}=1$

Take the inverse of tan to find the corresponding angle:

$\tan^{-1}(1)=45^{\circ}$

A short cut for solving is to realize right triangles with equal sides must have two 45° angles!

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Example Question #5 : Finding Sides

If the two legs of a right triangle are 5 and 7 respectively, what is the length of the hypotenuse?

Possible Answers:

$\sqrt{74}$

$\sqrt {80}$

$\sqrt {77}$

$\sqrt{73}$

Correct answer:

$\sqrt{74}$

Explanation:

Step 1: Recall the Pythagorean Theorem:

a^2+b^2=c^2 , where $a \and\ b$ are the legs of the triangle.

Step 2: Plug in the values that are given for the legs into the theorem:

(5)^2+(7)^2=c^2

Evaluate:

25+49=c^2

Add:

74=c^2

Step 3: To find , take the square root of both sides:

$\sqrt{74}=\sqrt{c^2}$

Reduce (if possible):

$\sqrt{74}=c$

The length of the hypotenuse is $\sqrt {74}$ .

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Example Question #6 : Finding Sides

In a right triangle, one of the legs has a length of and a hypotenuse of . Find the length of the other side...

Possible Answers:

Correct answer:

Explanation:

Step 1: To find the missing side of a right triangle using the Pythagorean Theorem:

a^2+b^2=c^2 .

Step 2: Substitute in the known values.

$\\a^2+b^2=c^2 \\8^2+b^2=17^2$

Step 3: Solve for the missing variable by manipulating the equation to isolate the variable.

$\\b=\sqrt{17^2-8^2} \\b=\sqrt{289-64} \\b=\sqrt{225} \\b=15$

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Example Question #1 : Finding Sides

Find the length of .

Possible Answers:

9.06

8.95

7.31

8.44

Correct answer:

8.44

Explanation:

Recall the Law of Sines for a generic triangle, as shown above:

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

Plug in the given values into the Law of Sines:

$\frac{14}{\sin 72}=\frac{b}{\sin 35}$

Rearrange the equation to solve for :

$b=(\sin 35)\frac{14}{\sin 72}=8.44$

Make sure to round to two places after the decimal.

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Example Question #1 : Finding Sides

Find the length of side .

Possible Answers:

2.73

5.11

1.90

3.08

Correct answer:

2.73

Explanation:

Recall the Law of Sines:

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

Plug in the given values into the Law of Sines:

$\frac{6}{\sin 85}=\frac{b}{\sin 27}$

Rearrange the equation to solve for :

$b=(\sin 27)\frac{6}{\sin 85}=2.73$

Make sure to round to two places after the decimal.

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Example Question #371 : Trigonometry

Find the length of side .

Possible Answers:

4.66

4.12

6.97

5.09

Correct answer:

4.66

Explanation:

Recall the Law of Sines:

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

Plug in the given values into the Law of Sines:

$\frac{7}{\sin 62}=\frac{b}{\sin 36}$

Rearrange the equation to solve for :

$b=(\sin 36)\frac{7}{\sin 62}=4.66$

Make sure to round to two places after the decimal.

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Example Question #11 : Finding Sides

Find the length of side .

Possible Answers:

7.26

8.59

9.17

9.38

Correct answer:

9.17

Explanation:

Recall the Law of Sines:

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

Plug in the given values into the Law of Sines:

$\frac{12}{\sin 81}=\frac{b}{\sin 49}$

Rearrange the equation to solve for :

$b=(\sin 49)\frac{12}{\sin 81}=9.17$

Make sure to round to two places after the decimal.

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Trigonometry : Solving Triangles

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #1 : Finding Sides

Example Question #2 : Finding Sides

Example Question #3 : Finding Sides

Example Question #4 : Finding Sides

Example Question #5 : Finding Sides

Example Question #6 : Finding Sides

Example Question #1 : Finding Sides

Example Question #1 : Finding Sides

Example Question #371 : Trigonometry

Example Question #11 : Finding Sides

All Trigonometry Resources

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