Tahat Just Math · Self-paced practice
Learn and Practice Right-Triangle Trigonometry
Build confidence with clear concepts, guided examples, and self-paced practice.
- 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.
541 questions · 10 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 4
Find missing sides using trigonometry and the Pythagorean theorem
Mark the given 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 trigonometry formulas below, replace \(A\) with \(B\) when angle B is given.
For the trigonometry formulas, \(o\) means opposite, \(a\) means adjacent, and \(h\) means hypotenuse relative to the chosen angle. These are side roles, not the fixed side labels \(a\), \(b\), and \(c\) used in the Pythagorean theorem.
Missing middle letter: multiply ( SOH, CAH, TOA )
If the missing side is the middle letter in the rule (the numerator), multiply the given side by the trigonometric ratio.
\(o=h\sin(A)\)
\(a=h\cos(A)\)
\(o=a\tan(A)\)
Missing last letter: divide ( SOH, CAH, TOA )
If the missing side is the last letter in the rule (the denominator), divide the given side by the trigonometric ratio.
\(h=\frac{o}{\sin(A)}\)
\(h=\frac{a}{\cos(A)}\)
\(a=\frac{o}{\tan(A)}\)
Two sides given: use \(a^2+b^2=c^2\)
In these formulas, \(a\) and \(b\) are the legs and \(c\) is the hypotenuse.
Add leg squares when finding the hypotenuse \(c\). Subtract the given leg square when finding a missing leg.
Find hypotenuse \(c=\sqrt{a^2+b^2}\)
Find leg a \(a=\sqrt{c^2-b^2}\)
Find leg b \(b=\sqrt{c^2-a^2}\)
Objective 3
Find missing angles with inverse trigonometry
Choose the inverse ratio that matches the given sides.
\(\theta=\sin^{-1}\left(\frac{o}{h}\right)\)
\(\theta=\cos^{-1}\left(\frac{a}{h}\right)\)
\(\theta=\tan^{-1}\left(\frac{o}{a}\right)\)
- Identify the given angle (A or B) and mark the given and missing sides.
- Choose the ratio (SOH–CAH–TOA) containing the given 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.