Technical Mathematics
Course Curriculum Directory
Choose an available chapter to open its interactive lessons, study tools, and practice questions.
Tahat Just Math
Chapter 13: Trigonometry & Right Triangles Practice · 427 questions
00:00
Tahat Just Math · Self-paced practice
Interactive Mathematics Simulation
Test questions, verify solutions step-by-step, and monitor live metrics.
- 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 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.
Objectives 1–3
Construct, calculate and interpret circle graphs
A circle graph shows how a whole is divided among categories. One full circle measures \(360^\circ\), so each category's central angle matches its share of the total.
Central angle from a percentage
\(\text{angle}=\frac{\text{percent}}{100}\times 360^\circ\)
angle = 25 / 100 × 360° = 90°
Central angle from a count
\(\text{angle}=\frac{\text{category count}}{\text{total count}}\times 360^\circ\)
Objectives 4–5
Read line graphs and bar graphs
Use a line graph to follow change over time. Use a bar graph to compare categories. Read the axis labels and scale before comparing values.
Line graph
Find the value on the horizontal axis, move to the line, then read across to the vertical axis.
Bar graph
Read each bar's height from the scale. Then add, subtract or compare as the question asks.
Objectives 6–7
Build frequency distributions and histograms
- Choose equal-width intervals that cover the full range of the data.
- Place each value in exactly one interval and count its frequency \(f\).
- Check that the frequencies add to the number of data values.
- Draw one bar per interval. In a histogram, neighboring bars touch.
Objectives 8–11
Mean, median and mode
Mean
\(\bar{x}=\frac{\sum x}{n}\)
Add the values, then divide by how many values there are.
Median
Put the data in order. With an even number of values, average the two middle values.
Mode
The value that appears most often. A data set can have one mode, more than one, or none.
Mean of a frequency distribution
\(\bar{x}=\frac{\sum xf}{\sum f}\)
Multiply each interval midpoint \(x\) by its frequency \(f\). Add those products and divide by the total frequency.
Interactive playground
Build a circle graph from a percentage
Change the percentage to see how its share of the circle and central angle change.
37% of the circle is \(133.2^\circ\).
Central angle = percentage \(\times 3.6^\circ\)
Try \(25\%\), \(50\%\) and \(75\%\). The category and the remainder should always add to \(100\%\).
Line graph: change the values
Move a slider to change one day's value and watch the line bend.
Bar graph: change the heights
Each slider sets the height of one bar. Compare the categories and their total.
Frequency distribution and histogram
Change the frequency of each class interval. The histogram bars touch, and the grouped mean uses the class midpoints.
Mean = \(\frac{\sum xf}{\sum f}\)
Mean, median and mode
Edit the data set (numbers separated by commas). The chart shows each value in order, with the mean and median marked.
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
Open a skill, read the idea, then work through one example a step at a time. Use each question’s hint and step-by-step solution controls in Practice problems 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 427 questions across 8 learning objectives. Choose any number of questions you like. A random set takes one question from each objective first, so a set of 8 or more questions covers the whole chapter.
Test ended
You left the test page 3 times. This attempt has ended and no score was recorded.
Fullscreen was exited
This departure is recorded. Return to fullscreen to continue the test.
Question 1 of 427
2% Completed
Loading question...
Section 1
Math notation could not be rendered. The original notation is still available in the question text.
✓
Correct Answer!
Review your answers
Unanswered
Flagged for review
Quiz Completed!
Review your score and choose whether to share it with Tahat Just Math.
0
out of 427
Time elapsed
00:00
0% Score
Final Grade: --
Save your test result
Download a copy of your result to keep or to give to your teacher.
Detailed Answers Review (Q1–Q427)
Show answers
Hide answers
Emailing your result opens a draft addressed to contact@tahatjustmath.com with your quiz and score (and your name, if you typed it above). Review the draft and send it yourself only if you want to share these details.