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## Trigonometry

### Unit 1: Lesson 6

Modeling with right triangles

# Right triangle trigonometry review

Review right triangle trigonometry and how to use it to solve problems.

## What are the basic trigonometric ratios?

A right triangle A B C where angle A C B is the right angle. Angle B A C is the angle of reference. Side A B is labeled hypotenuse. Side B C is labeled opposite. Side A C is labeled adjacent.
sine, left parenthesis, angle, A, right parenthesis, equalsstart fraction, start color #11accd, start text, o, p, p, o, s, i, t, e, end text, end color #11accd, divided by, start color #e07d10, start text, h, y, p, o, t, e, n, u, s, e, end text, end color #e07d10, end fraction
cosine, left parenthesis, angle, A, right parenthesis, equalsstart fraction, start color #aa87ff, start text, a, d, j, a, c, e, n, t, end text, end color #aa87ff, divided by, start color #e07d10, start text, h, y, p, o, t, e, n, u, s, e, end text, end color #e07d10, end fraction
tangent, left parenthesis, angle, A, right parenthesis, equalsstart fraction, start color #11accd, start text, o, p, p, o, s, i, t, e, end text, end color #11accd, divided by, start color #aa87ff, start text, a, d, j, a, c, e, n, t, end text, end color #aa87ff, end fraction

## Practice set 1: Solving for a side

Trigonometry can be used to find a missing side length in a right triangle. Let's find, for example, the measure of A, C in this triangle:
A right triangle A B C. Angle A C B is a right angle. Angle A B C is forty degrees. Side A C is unknown. Side A B is seven units.
We are given the measure of angle angle, B and the length of the start color #e07d10, start text, h, y, p, o, t, e, n, u, s, e, end text, end color #e07d10, and we are asked to find the side start color #11accd, start text, o, p, p, o, s, i, t, e, end text, end color #11accd to angle, B. The trigonometric ratio that contains both of those sides is the sine:
\begin{aligned} \sin(\angle B)&=\dfrac{\blueD{AC}}{\goldD{AB}} \\\\ \sin(40^\circ)&=\dfrac{AC}{7}\quad\gray{\angle B=40^\circ, AB=7} \\\\ 7\cdot\sin(40^\circ)&=AC \end{aligned}
Now we evaluate using the calculator and round:
A, C, equals, 7, dot, sine, left parenthesis, 40, degrees, right parenthesis, approximately equals, 4, point, 5
Problem 1.1
A right triangle A B C. Angle A C B is a right angle. Angle B A C is sixty-five degrees. Side B C is unknown. Side A B is six units.
B, C, equals

Want to try more problems like this? Check out this exercise.

## Practice set 2: Solving for an angle

Trigonometry can also be used to find missing angle measures. Let's find, for example, the measure of angle, A in this triangle:
A right triangle A B C. Angle A C B is a right angle. Angle B A C is unknown. Side A C is six units. Side A B is eight units.
We are given the length of the side start color #aa87ff, start text, a, d, j, a, c, e, n, t, end text, end color #aa87ff to the missing angle, and the length of the start color #e07d10, start text, h, y, p, o, t, e, n, u, s, e, end text, end color #e07d10. The trigonometric ratio that contains both of those sides is the cosine:
\begin{aligned} \cos(\angle A)&=\dfrac{\purpleC{AC}}{\goldD{AB}} \\\\ \cos(\angle A)&=\dfrac{6}{8}\quad\gray{AC=6, AB=8} \\\\ \angle A&=\cos^{-1}\left(\dfrac{6}{8}\right) \end{aligned}
Now we evaluate using the calculator and round:
angle, A, equals, cosine, start superscript, minus, 1, end superscript, left parenthesis, start fraction, 6, divided by, 8, end fraction, right parenthesis, approximately equals, 41, point, 41, degrees
Problem 2.1
A right triangle A B C. Angle A C B is a right angle. Angle B A C is unknown. Side B C is two units. Side A B is six units.
angle, A, equals
degrees

Want to try more problems like this? Check out this exercise.

## Practice set 3: Right triangle word problems

Problem 3.1
Howard is designing a chair swing ride. The swing ropes are 5 meters long, and in full swing they tilt in an angle of 29, degrees. Howard wants the chairs to be 2, point, 75 meters above the ground in full swing.
How tall should the pole of the swing ride be?
meters
The design of the chair swing ride. The pole of the swing is a rectangle with a short base and a long height. At the top of the pole, there are swing ropes that extend from the pole at an angle of twenty-nine degrees. The rope extends for 5 meters where there is a chair that is two point seventy-five meters off the ground.

Want to try more problems like this? Check out this exercise.

## Want to join the conversation?

• This is not correct. The path of the swing is an arc so at the point where it is parallel to the support pole it would closer to the ground than at the point of full swing which is 2.75 meters. To make this example correct the 2,75 meters needs to be applied to the point where the swing is parallel to the supporting pole.
• You are correct that it is an arc. However, the key to the question is the phrase "in full swing". The swing will be closer than 2.75 meters at the bottom of the arc. That is an interesting point that I hadn't considered, but not what the question is asking.
• Shouldn't we take in account the height at which the MIB shoots its laser. I'm guessing it would be somewhere from his shoulder. Maybe the answer wouldn't differ that much but it might make it a little more challenging to figure out. You would even be able to calculate the height the agent is holding his gun at with stretched arms when you know the angle he's keeping his arms at, his arm length and the length from his shoulders to the ground.
• Good point, let's estimate :D.

Men In Black are generally rather tall so it is fair to estimate the man is about two meters tall. The average arm length of an adult human is ~25 inches which equates to about 0.6 meters. If we assume that the man holds his arms directly above his head (not technically realistic but a fair assumption) then we can estimate the height of the LASER to be about 2.5 meters. If we subtract 2.5 from 324 we get 321.5. Arctan(321.5/54) = 80.465.

That deviates from the ground angle by only 0.09%, so it probably wouldn't affect how he aimed the laser.

If we assume he shot from shoulder height with his arms straight out, then that would be arctan(322/53.5) = ~80.567 which deviates from the ground angle by only 0.04%.
• What is the value of sine, cosine, and tangent?
• The Sine, Cosine, and Tangent are three different functions. They do not have a value outright, it would be like trying to ask what the value of f(x) = x + 1 is. The trig functions give outputs in terms of the ratios of two sides of a triangle when we feed them the input of an angle measure.
Sine outputs the ratio of opposite to hypotenuse
Cosine outputs the ratio of adjacent to hypotenuse
Tangent outputs the ratio of opposite to adjacent
• in question 1.1 the given answer is approx 5.44 my calculator is giving 0.91 as an answer even in degrees mode
• when working out the inverse trig, is the bigger number always on the bottom?
• For sine and cosine, yes because the hypotenuse will always be the longest side, but for tangent, it does not have to be, either the opposite or the adjacent could be longer than the other.
• do i have to be specific
• I agree with Spandan. If you aren't specific, because math has so many different terms, it's usually impossible to figure out exactly what you mean- there can be multiple answers to a question using or leaving out seemingly nonimportant words!
• My problem is that I do not know which one is adjacent and opposite you the one closest to the angle is adjacent but if it doesn't show the angle then how am I supposed to know which one.