Label the sides relative to the angle you are using, then pick the ratio containing what you know and what you want. Sine is opposite over hypotenuse, cosine adjacent over hypotenuse, tangent opposite over adjacent.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Try it first
Two right triangles stand on the same ground line and share the same angle at the bottom. The small one rises 3 m over a run of 4 m. The big one has that same bottom angle and a run of 20 m. How high does the big one rise, in meters?
meters
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
Label first
The hypotenuse is opposite the right angle and never changes. Opposite and adjacent depend on which acute angle you stand at. Swap angles and they swap names.
Pick the ratio
Know the hypotenuse and want the opposite: sine. Know the adjacent and want the hypotenuse: cosine. Know one leg and want the other: tangent. Two of the three sides name the ratio.
Finding a side
Hypotenuse 12, angle 40°, opposite wanted: sin 40° = x ÷ 12, so x = 12 × 0.643 = 7.7. When the unknown is on the bottom, multiply across and divide: cos 40° = 12 ÷ h gives h = 12 ÷ cos 40°.
Finding an angle
Legs 3 and 4, angle wanted: tan θ = 3 ÷ 4 = 0.75, so θ = tan⁻¹ 0.75 ≈ 36.9°. The inverse button turns a ratio back into an angle.
Elevation and depression
An angle of elevation is measured up from the horizontal; depression is measured down from it. The angle of elevation from A to B equals the angle of depression from B to A: they are alternate angles.
A sanity check
The hypotenuse is always the longest side, and the side opposite the smaller angle is the shorter leg. If your answer says otherwise, the ratio was upside down.
Step 2: Try It Yourself
Tap and try it out.
A 6 m ladder leans against a wall at 65° to the ground. How high does it reach, and how far is its foot from the wall?
- 1hypotenuse 6, angle 65°: height is opposite, foot distance is adjacentlabel from the 65° angle
Step 0 of 4
- sin — opposite over hypotenuse0.60
- cos — adjacent over hypotenuse0.80
- tan — opposite over adjacent0.75
- The marked angle36.87°
The hypotenuse is not given. It comes from 4² + 3² = 25, whose square root is 5.
Step 3: In Real Life
A surveyor’s sight line
From 100 meters away, a surveyor sights a tower top at 32 degrees. tan 32° × 100 gives about 62 meters. Every tall thing is measured by an angle and a distance.
Step 4: Watch an Example
One step at a time.
Watch Priya Find a Rope Length
A tent rope runs from a point 2.5 m up the pole down to a peg in the ground, meeting the ground at 55°. Priya wants the length of the rope.
- Step 1
From the 55° angle, where the rope meets the ground, the 2.5 m of pole is the side facing the angle, so it is the opposite. The rope itself runs from that angle to the top of the pole, across from the right angle at the foot: it is the hypotenuse.
Watch Dev Find an Angle of Depression
A lifeguard stands at the top of a 40 m cliff and looks down at a swimmer 50 m out from the foot of the cliff. Dev wants the angle of depression. He sets the first line down, you supply the next three, and he closes with the check.
- Step 1
The angle of depression is measured down from the horizontal at the lifeguard’s eye. That horizontal and the flat sea are parallel, so the angle of depression from the top equals the angle of elevation from the swimmer — alternate angles — and the swimmer’s corner sits inside a right triangle whose two legs are both known.
Two Roads to the Same Side
A right triangle has a hypotenuse of 20 m and an acute angle of 35°. The side lying alongside that angle is wanted, and there are two honest ways to reach it.
- Step 1
A hypotenuse and an angle are given, and the side alongside the angle is wanted: adjacent and hypotenuse, which is the cosine’s pair.
Step 5: Your Turn
Practice makes it stick.
The Guy Wire
Problem 1 of 2
A mast is held upright by a wire running from its top down to an anchor in the ground. The wire meets the ground at 60°, and the anchor is 14 m from the foot of the mast. Wire costs $12 a meter. What does the wire cost, in dollars?
dollars
The Tree Survey
Problem 2 of 2
A tree surgeon prices a job by height: $22 for every meter of tree. Standing 30 m from the trunk on level ground, she sights the top at an angle of elevation of 45°, measured from her eye, which is 2 m above the ground. What is the bill, in dollars?
dollars
Solve the Triangle
Label the sides from the angle named, then pick the ratio holding what you know and what you want. Ratios to three decimals, answers to one.
1 of 5
A right triangle has a hypotenuse of 18 cm and an acute angle of 35°. How long is the side opposite that angle, in centimeters to one decimal place?
cm
2 of 5
A right triangle has an acute angle of 55°, and the side alongside that angle is 9 cm. How long is the hypotenuse, in centimeters to one decimal place?
cm
3 of 5
A right triangle has legs of 9 cm and 40 cm. What is the angle between the 40 cm leg and the hypotenuse, in degrees to one decimal place?
degrees
4 of 5
A ramp rises 2 m over a sloping length of 8 m. What angle does the ramp make with the ground, in degrees to one decimal place?
degrees
5 of 5
A right triangle has a hypotenuse of 25 cm and one leg of 7 cm. What is the angle between the hypotenuse and that 7 cm leg, in degrees to one decimal place?
degrees
Whose Step?
One line in each page is a slip. The reasons are not printed: find it from the lines and the margins, then fix it.
1 of 6
A kite string runs from a hand at ground level up to a kite 30 m above the ground, meeting the ground at 40°. Maya found the string 19.3 m long. One line is a slip. Which?
Maya’s page
- 1angle 40°: the 30 m of height is opposite, the string L is the hypotenuse
- 2sin 40° = 30 ÷ L
- 3L = 30 × sin 40° = 30 × 0.643× sin 40°
- AnswerL ≈ 19.3 m
2 of 6
Fix Maya’s line 3 and finish her page. How long is the kite string, in meters to one decimal place?
meters
3 of 6
A ladder’s foot stands 2 m out from a wall and the ladder meets the ground at 70°. Leo found that it reaches 0.7 m up the wall. One line is a slip. Which?
Leo’s page
- 1angle 70°: the 2 m to the wall is opposite, the height h is adjacent
- 2tan 70° = 2 ÷ h
- 3h = 2 ÷ tan 70° = 2 ÷ 2.747÷ tan 70°
- Answerh ≈ 0.7 m
4 of 6
Label Leo’s triangle the other way round and finish. How high up the wall does the ladder reach, in meters to one decimal place?
meters
5 of 6
A slide drops 3.5 m over a sloping length of 5 m. Ravi found that it makes an angle of 0.7° with the ground. One line is a slip. Which?
Ravi’s page
- 1the 5 m slope is the hypotenuse, the 3.5 m drop is opposite
- 2sin θ = 3.5 ÷ 5 = 0.7
- Answerθ ≈ 0.7°
6 of 6
Fix Ravi’s last line. What angle does the slide make with the ground, in degrees to one decimal place?
degrees
Pick the Move
Choose the first line only. Nothing here needs working out.
1 of 2
A right triangle has a hypotenuse of 20 m and an acute angle of 28°. The side lying alongside that angle is wanted, and x is its length. Which first line does the job?
2 of 2
A right triangle has legs of 6 cm and 11 cm. The angle between the 11 cm leg and the hypotenuse is wanted, and θ is that angle. Which first line does the job?
Same Ratio?
Nothing here needs a calculator. Reason from which corner of the triangle you are standing at.
1 of 2
In a right triangle the two acute angles are 40° and 50°. Side a faces the 40° angle, side b faces the 50° angle, and c is the hypotenuse. Select every expression that has the same value as sin 40°.
2 of 2
A right triangle has an acute angle of 65°. Select every statement that must be true, without working anything out.
Step 6: Quick Check
Show what you know.
Question 1 of 2
Legs 20 and 21. Hypotenuse?
Question 2 of 2
Which ratio involves the opposite and adjacent sides?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
A surveyor on level ground sights the top of a tower at an angle of elevation of 30°. She walks 40 m straight towards the tower and sights the top again, now at 45°. Neither distance to the tower was measured. How tall is the tower, in meters to one decimal place?
meters
Stretch 2 of 2
A ramp has to rise exactly 1 m to reach a doorway, and a safety rule says it may meet the ground at no more than 8°. Space is tight, so the shortest ramp that obeys the rule is wanted. How long is its sloping surface, in meters to one decimal place?
meters
What You Learned
- Label the sides relative to your chosen angle.
- Pick the ratio containing what you know and what you want.
- An inverse ratio turns a ratio back into an angle.
- The hypotenuse is always the longest side; check your answer against it.
For the grown-up
The misconception this lesson targets is that SOH-CAH-TOA is a spelling rather than a choice. A student can recite the three ratios perfectly and still be stuck, because opposite and adjacent mean nothing until you have decided which corner you are standing at — and the commonest slip here is a side labelled from the wrong corner, not a ratio misremembered. Its close cousin is the unknown that sits underneath: sin 40° = 30 ÷ L is freed by swapping L with sin 40°, never by multiplying by it. Both slips announce themselves the same way, in an answer that is the wrong size — a hypotenuse shorter than one of its own legs, or an angle of well under a degree where the picture is plainly steep. Asking "is that the right size?" catches more in this lesson than re-checking the arithmetic does, and it is the habit worth insisting on.