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.
Prerequisite skills
Refresh the tools behind trigonometry
Pythagorean calculator
Find a missing side in a right triangle. Enter positive lengths; the hypotenuse must be longer than either leg.
Quick refresher: Pythagorean Theorem
In a right triangle, the hypotenuse c is opposite the right angle. If a leg is missing, isolate it before taking a square root.
\(a^2+b^2=c^2\), so \(a=\sqrt{c^2-b^2}\).
Complementary angles
The two acute angles of a right triangle add to \(90^\circ\).
\(m\angle A + m\angle B = 90^\circ\)
If \(m\angle A=34^\circ\), then \(m\angle B=90^\circ-34^\circ=56^\circ\).
Types of triangles
Triangles can be classified by their side lengths or by their angles.
By side lengths
EquilateralAll three sides are equal.
IsoscelesAt least two sides are equal.
ScaleneAll three sides have different lengths.
By angles
AcuteAll three angles are less than 90°.
RightOne angle is exactly 90°.
ObtuseOne angle is greater than 90°.
Side and angle classifications describe different features. For example, a triangle can be both isosceles and right.
Triangle anatomy and SOH–CAH–TOA
The hypotenuse stays the same. Opposite and adjacent depend on which acute angle you choose.
Reference angle
HypotenuseOpposite 90° · side c
OppositeAcross from angle A · side a
AdjacentTouches angle A · side b
Angle A selected: side a is opposite and side b is adjacent.
SOH · Sine
CAH · Cosine
TOA · Tangent
Use the ratio that contains the side you know and the side you need.
Interactive playground
Explore how a right triangle changes
Adjust either leg to see the hypotenuse, reference angle, and trigonometric ratios update.
Hypotenuse \(c\)10.00
Angle A36.87°
sin(A) = a/c0.6000
cos(A) = b/c0.8000
tan(A) = a/b0.7500
Exact trigonometric values
Special right triangles
Use side patterns to find exact ratios without a calculator.
Exact ratios for special right triangles
Recognize these side patterns to find exact values without a calculator.
45°–45°–90° · Isosceles right triangle
Both legs have the same length.
Side ratio: \(x:x:x\sqrt{2}\)
30°–60°–90° · Half of an equilateral triangle
The short leg is opposite 30°; the hypotenuse is twice the short leg.
Side ratio: \(x:x\sqrt{3}:2x\)
| Ratio | 30° | 45° | 60° |
|---|---|---|---|
| Sine | \(\frac{1}{2}\) | \(\frac{\sqrt{2}}{2}\) | \(\frac{\sqrt{3}}{2}\) |
| Cosine | \(\frac{\sqrt{3}}{2}\) | \(\frac{\sqrt{2}}{2}\) | \(\frac{1}{2}\) |
| Tangent | \(\frac{\sqrt{3}}{3}\) | \(1\) | \(\sqrt{3}\) |
Learn by doing
Guided examples by skill
Choose an objective to open its question set. Use each question’s hint and step-by-step solution controls as needed.
Scientific calculator
Practice calculator (fx-82 style)
Press SHIFT or ALPHA first to reach the gold or red labels above the keys. Use MODE for DEG/RAD/GRA, rounding and statistics. This is an original practice calculator, not a Casio product.
Check your work
Trigonometry calculator
Choose a ratio, enter two known values, and solve for an angle or side. Angles are in degrees.
Explore a right triangle
Remember before calculating
Enter any two positive side lengths and leave the third blank. The calculator uses the Pythagorean theorem, then reports the selected ratio and the reference angle.
Check degree modeFor angle problems in degrees, confirm the calculator shows DEG before using a trig or inverse-trig key.
Keep the reference angle in viewOpposite and adjacent change when you switch angles; the hypotenuse is always opposite the right angle.
Round at the endKeep extra digits in intermediate steps, then round the final value as requested.
Model the situation firstDraw the horizontal reference line and label known sides before choosing a ratio.
Calculator checks and common pitfalls
Use exact values when possible and round only when the question asks for an approximation.
Chapter 13 · Mock exam
A focused 20-question test
Questions are selected at random from the objectives you choose, with at least one question from each selected objective. Answer in any order, flag questions to revisit, and review everything before you submit. Your score and answer review appear at the end. Use your school-approved tablet; phones should remain put away under classroom rules.
- 20random questions
- 40 minutesauto-submits when time ends
- Tab & focus monitoringends after 3 departures
Switching tabs or focusing another window is counted. Some split-screen modes keep both pages focused and visible, which a website cannot detect. This is not secure proctoring.
The test requests fullscreen when you start. Exiting fullscreen is recorded and pauses the test until you return to fullscreen. If fullscreen is unavailable, you can choose to continue without it.
Focused practice
Choose a learning objective
Practice one skill at a time, or use the full assessment from the hero section to review the entire chapter.
Random practice
Mixed practice across every objective
Chapter 13 contains 541 questions across 10 learning objectives. A random set always includes at least one question from every objective, so a set of 10 questions covers the whole chapter.