Flip the ratios around. When you know two sides, inverse sine, cosine and tangent reveal the hidden angle.
Today's hook: A builder needs to know the pitch of a roof before ordering tiles. They can't climb up and measure the angle directly -- but they can measure the base and height from the ground, then work backwards through the ratio.
0/5QUESTS
Printable Worksheets
Print or save as PDF — or build a custom worksheet from any module's questions.
In a right-angled triangle, the opposite side is 5 cm and the hypotenuse is 10 cm. How would you find the angle? What calculator button would you press?
Record your answer in your workbook.
1
The Big Idea
+5 XP to read
When you know two sides of a right-angled triangle, you can work backwards through the ratio to find the angle. The inverse functions $\sin^{-1}$, $\cos^{-1}$ and $\tan^{-1}$ are the undo buttons of trigonometry.
If $\sin \theta = 0.5$, then $\theta = \sin^{-1}(0.5) = 30°$. The inverse function reads the ratio and returns the angle. Every trig function has an inverse that walks the calculation backwards.
You cannot find an angle with only one side. You need a pair.
Answer is an angle
The output of $\sin^{-1}$ is always degrees (or radians). Not a length.
2
What You'll Master
objectives
Know
The inverse trig functions $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$
That calculators must be in DEGREE mode
How to round angles to the nearest degree
Understand
Why inverse trig functions undo the original trig functions
How to choose the correct inverse function based on known sides
Can Do
Find an unknown angle given two sides of a right-angled triangle
Set up the correct ratio before using the inverse function
Round answers and verify they make sense
3
Words You Need
vocabulary
Inverse sine ($\sin^{-1}$)The angle whose sine is a given value. Also written as arcsin.
Inverse cosine ($\cos^{-1}$)The angle whose cosine is a given value. Also written as arccos.
Inverse tangent ($\tan^{-1}$)The angle whose tangent is a given value. Also written as arctan.
DEG modeCalculator setting where angles are measured in degrees, not radians.
RatioThe fraction formed by dividing one side by another. The input for an inverse function.
VerifyTo check an answer by substituting it back into the original problem.
4
Spot the Trap
heads-up
Wrong: Using $\sin^{-1}$ when you know opposite and adjacent. $\sin^{-1}$ is only for opposite/hypotenuse pairs.
Right: Match the two sides you know to SOH CAH TOA, then use the corresponding inverse function.
Wrong: Calculator in RAD mode gives answers like 0.524 instead of 30°.
Right: Always check your calculator shows DEG or D before calculating angles.
5
Parts of the Whole
+5 XP
Every angle-finding problem has the same three ingredients: a marked angle you need to find, two sides whose lengths you already know, and the inverse function that connects them.
Before you touch your calculator, identify the unknown angle, the two known sides, and which inverse function matches the pair. These three pieces determine everything else.
$\theta = \sin^{-1}\left(\frac{O}{H}\right)$
Name the unknown angle
Call it $\theta$ so you know what you are solving for.
Spot the two known sides
You need exactly two side lengths to find an angle.
Pick the matching inverse
O+H = sin−1, A+H = cos−1, O+A = tan−1.
6
Flip the Ratio
+5 XP
The inverse functions walk backwards. If sine turns an angle into a ratio, then inverse sine turns a ratio back into an angle. Each trig function has an undo button.
$\sin^{-1}$ undoes sine. $\cos^{-1}$ undoes cosine. $\tan^{-1}$ undoes tangent. Write the ratio first, then press the matching inverse button on your calculator.
$\sin^{-1}(0.5) = 30°$
sin−1 undoes sine
If sin turns 30° into 0.5, sin−1 turns 0.5 back into 30°.
cos−1 undoes cosine
The angle whose cosine equals your ratio.
tan−1 undoes tangent
The angle whose tangent equals your ratio.
7
Follow the Steps
+5 XP
Six steps, every time. Master this sequence and every angle-finding problem becomes automatic.
Label the sides. Identify the two known sides. Choose the ratio. Write the ratio as a fraction or decimal. Apply the inverse function. Round and verify.
$\theta = \tan^{-1}\left(\frac{12}{5}\right)$
Write the ratio first
Never reach for the calculator until the ratio is written down.
Calculate the decimal
$\frac{7}{14} = 0.5$. Then press sin−1(0.5).
Round at the end
Keep full precision during calculation. Round only the final answer.
8
Check and Verify
+5 XP
After finding the angle, check three things: your calculator was in DEG mode, the angle is reasonable, and substituting back gives the original ratio.
A quick sanity check catches most errors. If your ratio is 0.5, the angle should be around 30°, not 0.5°. If you got 0.524, your calculator is in RAD mode. Always verify by substituting back.
$30° \rightarrow \sin 30° = 0.5$ ✓
DEG not RAD
0.524 is radians. 30° is degrees. Check the screen.
Small ratio = small angle
A ratio near 0 means a small angle. A ratio near 1 means a large angle.
Verify by substitution
Plug your angle back in. You should get the original ratio.
Watch Me Solve It · 3 examples
Watch Me Solve It · Finding an angle
+15 XP per step
Q1
PROBLEM
In a right-angled triangle, opposite = 7 and hypotenuse = 14. Find $\theta$ to the nearest degree.
1
Label the sides
Opposite = 7, Hypotenuse = 14, Angle = $\theta$
Identify which sides are involved before choosing a ratio.
2
Choose the ratio
We have Opposite and Hypotenuse → use sine
Opposite + Hypotenuse pairs with sine (SOH).
3
Write and simplify
$\sin \theta = \frac{7}{14} = 0.5$
4
Apply the inverse function
$\theta = \sin^{-1}(0.5) = 30°$
Check DEG mode, then press sin−1(0.5).
Nice work -- XP earned
Answer$\theta = 30°$
Watch Me Solve It · Using tangent
+15 XP per step
Q2
PROBLEM
In a right-angled triangle, opposite = 12 and adjacent = 5. Find $\theta$ to 1 decimal place.
1
Label the sides
Opposite = 12, Adjacent = 5, Angle = $\theta$
2
Choose the ratio
We have Opposite and Adjacent → use tangent
Opposite + Adjacent pairs with tangent (TOA).
3
Write and simplify
$\tan \theta = \frac{12}{5} = 2.4$
4
Apply the inverse and round
$\theta = \tan^{-1}(2.4) = 67.38°$ → 67.4°
$\tan 67° \approx 2.36$ and $\tan 68° \approx 2.48$, so 67.4° is reasonable.
Nice work -- XP earned
Answer$\theta = 67.4°$
Watch Me Solve It · Wheelchair ramp angle
+15 XP per step
Q3
PROBLEM
A ramp is 8 m long and rises 1.5 m vertically. Find the angle the ramp makes with the horizontal to the nearest degree.
1
Label the sides
Hypotenuse = 8 m (ramp), Opposite = 1.5 m (rise), Angle = $\theta$
2
Choose the ratio
We have Opposite and Hypotenuse → use sine
3
Write and simplify
$\sin \theta = \frac{1.5}{8} = 0.1875$
4
Apply the inverse and round
$\theta = \sin^{-1}(0.1875) = 10.81°$ → 11°
A shallow rise over a long ramp gives a small angle -- the answer makes sense.
Nice work -- XP earned
Answer$\theta = 11°$
9
Common Pitfalls
heads-up
Choosing the wrong inverse function
Knowing opposite and adjacent but pressing sin−1 because "it feels right". Always match the pair of sides to SOH / CAH / TOA first.
Fix: say aloud "I have Opposite and Adjacent -- that is tan -- so I need tan−1."
Calculator in RAD mode
A calculator in RAD mode makes sin−1(0.5) = 0.524 instead of 30°. Every answer is completely wrong.
Fix: look for D or DEG in the top corner of the screen before every problem.
Not verifying the answer
Writing down 67.4° without checking if tan(67.4°) is close to 2.4. Verification catches calculator-mode errors instantly.
Fix: after finding the angle, substitute it back into the original ratio as a sanity check.
Copy Into Your Books
Inverse Functions
$\sin^{-1}$ undoes sine
$\cos^{-1}$ undoes cosine
$\tan^{-1}$ undoes tangent
The Six Steps
1. Label -- name O, A, H
2. Identify -- the two known sides
3. Choose -- SOH / CAH / TOA
4. Write -- the ratio as a decimal
5. Inverse -- apply sin−1 / cos−1 / tan−1
6. Round -- and verify
Calculator Check
Confirm DEG mode
Small ratio = small angle
Verify by substitution
Reasonableness
Ratio near 0 → angle near 0°
Ratio near 1 → angle near 90°
Ratio = 1 → angle = 45°
How are you completing this lesson?
Brain Trainer · 4 problems
D
Brain Trainer · Mixed
4 problems
Four problems finding unknown angles. Work each one, then reveal the answer.
1 Find $\theta$ when opposite = 9 and hypotenuse = 15.
Explanation: The inverse sine function returns the angle whose sine equals the given value. Since $\sin 30° = 0.5$, we know $\sin^{-1}(0.5) = 30°$.
MC2
Find the angle
+10 XP
In a right-angled triangle, the opposite side is 8 and the adjacent side is 6. Find $\theta$ to 1 decimal place.
Correct -- tan−1(8/6) = 53.1°.
Not quite -- you have Opposite and Adjacent, so use tan−1.
Explanation: Opposite + Adjacent means tangent (TOA). $\tan \theta = \frac{8}{6} = 1.333$, so $\theta = \tan^{-1}(1.333) = 53.1°$.
MC3
Calculator mode trap
+10 XP
Your calculator is in RAD mode. You press $\sin^{-1}(0.5)$. What do you get?
Correct -- RAD mode gives 0.524 radians, not 30 degrees.
Watch out -- in RAD mode, inverse trig answers are in radians.
Explanation: In RAD mode, $\sin^{-1}(0.5) = \frac{\pi}{6} \approx 0.524$ radians. In DEG mode, it is 30°. Always check your calculator display.
MC4
Matching sides to function
+10 XP
You know adjacent = 9 and hypotenuse = 15. Which inverse function do you use?
Correct -- Adjacent + Hypotenuse = cosine (CAH).
Check SOH CAH TOA -- adjacent and hypotenuse pair with cosine.
Explanation: Adjacent and Hypotenuse are linked by cosine ($\cos \theta = \frac{A}{H}$), so use $\cos^{-1}$ to find the angle.
MC5
Special angle recognition
+10 XP
If $\tan \theta = 1$, what is $\theta$?
Correct -- $\tan 45° = 1$ exactly.
Think about when opposite equals adjacent -- a 45° isosceles triangle.
Explanation: A right-angled triangle where opposite = adjacent is a 45-45-90 triangle. $\tan 45° = \frac{1}{1} = 1$.
Short Answer · 3 questions
Q6
Find the angle, hypotenuse and verify
+15 XP
Q6
SHORT ANSWER
In a right-angled triangle, the opposite side is 7 and the adjacent side is 24. Find $\theta$ to the nearest degree. Then find the hypotenuse using Pythagoras' theorem. Finally, verify your angle by checking that $\sin \theta = \frac{7}{25}$.
Verify: $\sin 16° = 0.276$, and $\frac{7}{25} = 0.28$. Close enough -- rounding is the reason for the small difference.
Q7
Both acute angles in a 5-12-13 triangle
+15 XP
Q7
SHORT ANSWER
A right-angled triangle has sides 5, 12 and 13 (where 13 is the hypotenuse). Find both acute angles to 1 decimal place. Verify your answers by checking that they sum to 90°.
Write your working in your book.
Label one angle $\theta$. Opposite = 5, Hypotenuse = 13.
A wheelchair ramp rises 1.2 m over a horizontal distance of 8 m. Find the angle of the ramp to the nearest degree. Australian accessibility standards require the angle to be no more than 4.76°. Does this ramp comply? If not, by how many degrees does it exceed the standard?
Write your working in your book.
Label: Opposite = 1.2 m (rise), Adjacent = 8 m (horizontal), Angle = $\theta$.
Choose: Opposite + Adjacent = tangent.
Write: $\tan \theta = \frac{1.2}{8} = 0.15$.
Inverse: $\theta = \tan^{-1}(0.15) = 8.53°$ → 9° to the nearest degree.
Check compliance: 8.53° > 4.76° -- this ramp does not comply.
It exceeds the standard by $8.53° - 4.76° = 3.77°$ → approximately 4°.
S
Stretch Challenge · Ladder slip
+20 XP
S
STRETCH
A 5 m ladder leans against a wall. The base of the ladder is 2 m from the wall. Find the angle the ladder makes with the ground to 1 decimal place. The ladder slips so the base is now 3 m from the wall. Find the new angle. By how many degrees did the angle decrease?
Record in your book -- full marks require clear working.
Use the interactive below to test different side combinations. Input two sides and see how the triangle and inverse function change. Try to predict the angle before pressing the button.
Record two observations about how the inverse function changes with different side pairs.
Consolidation Game -- Trig Ratio Blaster
+10 XP for playing
Blast the correct answers before they reach the bottom. This game tests your speed at choosing the right inverse function for given side pairs.
Lesson Complete
+10 XP
Mark this lesson as complete to earn your bonus XP.