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Tahat Just Math
Chapter 13: Trigonometry & Right Triangles Question Bank · 427 questions
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- Identify hypotenuse, opposite, and adjacent sides.
- Use Pythagorean theorem & simplify radicals.
- Apply SOH–CAH–TOA and inverse ratios.
- Model real situations with angles of elevation/depression.
427 questions · 8 learning objectives
Objective 1
Identify sides relative to a reference angle \(\theta\)
Every right triangle has three sides. The hypotenuse is fixed by the right angle; opposite and adjacent are named relative to the chosen acute angle.
Opposite
Directly across from the reference angle \(\theta\).Adjacent
Touches the reference angle \(\theta\), but is not the hypotenuse.
Objective 2
Use the primary trigonometric ratios (SOH–CAH–TOA)
SOH · Sine
\(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\)
CAH · Cosine
\(\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\)
TOA · Tangent
\(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\)
Objective 3
Find unknown angles with inverse trigonometry
When the side lengths are known, choose the inverse ratio that matches the given sides.
\(\theta=\sin^{-1}\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)\)
\(\theta=\cos^{-1}\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right)\)
\(\theta=\tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right)\)
Objective 4
Find missing sides using trigonometry and the Pythagorean theorem
First mark the known angle and label the sides opposite, adjacent, and hypotenuse. With an angle and one side, use SOH–CAH–TOA. With two side lengths, use the Pythagorean theorem. In the formulas below, replace \(A\) with \(B\) when angle B is given.
Missing middle letter: multiply ( SOH, CAH, TOA )
If the missing side is the middle letter in the rule, multiply the known side by the trig ratio.
\(\text{opposite}=\text{hypotenuse}\cdot\sin(A)\)
\(\text{adjacent}=\text{hypotenuse}\cdot\cos(A)\)
\(\text{opposite}=\text{adjacent}\cdot\tan(A)\)
Missing last letter: divide ( SOH, CAH, TOA )
If the missing side is the last letter in the rule, divide the known side by the trig ratio.
\(\text{hypotenuse}=\frac{\text{opposite}}{\sin(A)}\)
\(\text{hypotenuse}=\frac{\text{adjacent}}{\cos(A)}\)
\(\text{adjacent}=\frac{\text{opposite}}{\tan(A)}\)
Two sides known: use \(a^2+b^2=c^2\)
Add leg squares when finding the hypotenuse \(c\). Subtract the known leg square when finding a missing leg.
\(c=\sqrt{a^2+b^2}\)
\(a=\sqrt{c^2-b^2}\)
\(b=\sqrt{c^2-a^2}\)
- Identify the given angle (A or B) and mark each known and unknown side.
- Choose the ratio (SOH–CAH–TOA) containing the known side and the missing side.
- Rearrange first: multiply if the missing side is the middle letter; divide if it is the last letter.
- Substitute, calculate in degree mode, and round only the final answer.
Objective 5
Apply trigonometry to real-world situations
Angle of elevation
The angle measured upward from a horizontal line of sight to a point above the observer.
For height \(h\) and horizontal distance \(d\): \(h=d\tan(\theta)\).
Angle of depression
The angle measured downward from a horizontal line of sight to a point below the observer.
Use the equal interior angle at the base of the triangle to choose a trigonometric ratio.