Ratios of a right triangle: sin, cos, tan
When a ladder is leaned on a wall, changing its angle changes how high its top reaches. This link between an angle and the ratios of the sides is trigonometry.
In a right triangle the hypotenuse is the longest side. For the chosen angle θ, the side facing it is the opposite and the side touching it (other than the hypotenuse) is the adjacent. sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, tan θ = opposite ÷ adjacent.
In the figure: Sides 3, 4, 5: for θ, sin = 3/5, cos = 4/5, tan = 3/4.
Ratios of a right triangle: sin, cos, tan
In the app you can move this figure with your finger.
Open in the appRule 1: Identify opposite and adjacent first — If the angle changes, the opposite and adjacent sides swap. The hypotenuse is always the side facing the right angle.
- Easy: Sides 3, 4, 5: for θ, sin = 3/5, cos = 4/5, tan = 3/4.
- Medium: In the same triangle for the other angle (opposite 4, adjacent 3): sin = 4/5, cos = 3/5, tan = 4/3.
- Exam type: 5, 12, 13: for the larger angle sin = 12/13, cos = 5/13, tan = 12/5; for the smaller angle sin = 5/13 and cos = 12/13. The sin of one angle is the cos of the other.
Rule 2: Given one ratio, find the third side by Pythagoras — sin θ = 8/17 means opposite 8 and hypotenuse 17. Adjacent = √(17² − 8²) = 15. Now cos and tan follow easily.
- Easy: sin θ = 3/5 ⇒ adjacent 4 ⇒ cos θ = 4/5.
- Medium: cos θ = 5/13 ⇒ opposite 12 ⇒ sin θ = 12/13, tan θ = 12/5.
- Exam type: tan θ = 7/24 ⇒ hypotenuse √(49 + 576) = 25 ⇒ sin θ = 7/25, cos θ = 24/25.
Rule 3: The ratio depends on the angle, not on the triangle size — In similar triangles the sides grow but the ratio stays the same. In 6, 8, 10 also sin = 6/10 = 3/5.
- Easy: 3, 4, 5 and 6, 8, 10: tan is 3/4 in both.
- Medium: In a triangle with opposite 9 and hypotenuse 15: sin = 9/15 = 3/5.
- Exam type: In the same triangle adjacent = √(225 − 81) = 12, cos = 12/15 = 4/5, tan = 9/12 = 3/4.
sin = opposite ÷ hypotenuse, cos = adjacent ÷ hypotenuse, tan = opposite ÷ adjacent. The value of sin or cos is never more than 1.
Swapping opposite and adjacent, or finding sin/cos without using the hypotenuse.
Chant: "opp-hyp", "adj-hyp", "opp-adj" (sin, cos, tan). Memory clue: SOH-CAH-TOA: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent.
Spot the difference
| Ratio | Formula | In 3-4-5 (θ faces 3) |
|---|---|---|
| sin θ | opposite ÷ hypotenuse | 3/5 |
| cos θ | adjacent ÷ hypotenuse | 4/5 |
| tan θ | opposite ÷ adjacent | 3/4 |
Practice questions
- 1In the 5, 12, 13 triangle find sin of the angle facing 12.
- 2What is its cos?
- 3What is its tan?
- 4If sin θ = 4/5, find cos θ.
- 5If cos θ = 8/17, find sin θ.
- 6In a triangle the opposite is 6 and the adjacent is 8; find tan and sin.
- 7If tan θ = 5/12, find sin θ and cos θ.
- 8The opposite is 20 and the hypotenuse is 25. Find the adjacent, cos and tan.
Show answers (try them yourself first)
- 112/13. 2) 5/13. 3) 12/5.
- 4Opposite 4, hypotenuse 5 ⇒ adjacent 3 ⇒ cos θ = 3/5. 5) Adjacent 8, hypotenuse 17 ⇒ opposite 15 ⇒ sin θ = 15/17. 6) Hypotenuse 10. tan = 6/8 = 3/4 and sin = 6/10 = 3/5.
- 7Hypotenuse 13. sin θ = 5/13, cos θ = 12/13. 8) Adjacent √(625 − 400) = 15. cos = 15/25 = 3/5 and tan = 20/15 = 4/3.
Reciprocal ratios and identities
A coin has a reverse side, and each ratio has a reverse partner. The reverse of opposite ÷ hypotenuse is hypotenuse ÷ opposite.
Three reciprocal ratios: cosec θ = 1 ÷ sin θ, sec θ = 1 ÷ cos θ, cot θ = 1 ÷ tan θ. Two important identities: tan θ = sin θ ÷ cos θ and sin²θ + cos²θ = 1.
In the figure: The pairs sin-cosec, cos-sec and tan-cot are reciprocals of each other.
Rule 1: Reciprocal means flip the fraction — sin θ = 3/5 ⇒ cosec θ = 5/3. One gives the other at once.
- Easy: sin θ = 3/5 ⇒ cosec θ = 5/3.
- Medium: cos θ = 4/5 ⇒ sec θ = 5/4; tan θ = 3/4 ⇒ cot θ = 4/3.
- Exam type: If sin θ = 5/13 and cos θ = 12/13 then cosec θ = 13/5, sec θ = 13/12, cot θ = 12/5.
Rule 2: tan θ = sin θ ÷ cos θ — Both ratios are divided by the hypotenuse, so it cancels and opposite ÷ adjacent remains.
- Easy: sin = 3/5, cos = 4/5 ⇒ tan = (3/5) ÷ (4/5) = 3/4.
- Medium: sin = 12/13, cos = 5/13 ⇒ tan = 12/5.
- Exam type: sin = 8/17, cos = 15/17 ⇒ tan = 8/15 and cot = 15/8.
Rule 3: sin²θ + cos²θ = 1 — sin² means the square of sin. This is Pythagoras written another way. If one is known the other can be found.
- Easy: sin = 3/5: (3/5)² + (4/5)² = 9/25 + 16/25 = 1.
- Medium: sin = 5/13 ⇒ cos² = 1 − 25/169 = 144/169 ⇒ cos = 12/13.
- Exam type: cos = 7/25 ⇒ sin² = 1 − 49/625 = 576/625 ⇒ sin = 24/25 ⇒ tan = 24/7.
cosec, sec, cot are the reciprocals of sin, cos, tan. sin² + cos² = 1 is the most important identity.
Thinking cosec is the reciprocal of cos. Cosec is the reciprocal of sin; sec is the reciprocal of cos.
Cross-check: the first letters do not match: cosec ↔ sin and sec ↔ cos. Memory clue: Reciprocals: cosec ↔ sin, sec ↔ cos, cot ↔ tan; if one has "co", its partner does not.
Spot the difference
| Ratio | Reciprocal | Formula |
|---|---|---|
| sin θ | cosec θ | 1 ÷ sin θ |
| cos θ | sec θ | 1 ÷ cos θ |
| tan θ | cot θ | 1 ÷ tan θ |
Practice questions
- 1If sin θ = 4/5, what is cosec θ?
- 2If cos θ = 12/13, what is sec θ?
- 3If tan θ = 2/3, what is cot θ?
- 4From sin = 3/5, cos = 4/5 find cot θ.
- 5If sin θ = 12/13, find cos θ.
- 6If cos θ = 3/5, find sin θ and tan θ.
- 7If sin θ = 7/25, find cos θ, tan θ and sec θ.
- 8If cosec θ = 17/8, find sin θ, cos θ and tan θ.
Show answers (try them yourself first)
- 1cosec θ = 5/4. 2) sec θ = 13/12. 3) cot θ = 3/2.
- 4tan = 3/4 ⇒ cot = 4/3. 5) cos² = 1 − 144/169 = 25/169 ⇒ cos = 5/13. 6) sin² = 1 − 9/25 = 16/25 ⇒ sin = 4/5; tan = 4/3.
- 7cos² = 1 − 49/625 = 576/625 ⇒ cos = 24/25; tan = 7/24; sec = 25/24. 8) sin = 8/17 ⇒ cos² = 1 − 64/289 = 225/289 ⇒ cos = 15/17; tan = 8/15.
Values for 30°, 45° and 60°
Keep the two set squares (45°-45°-90° and 30°-60°-90°) in hand. Their side ratios are always the same, so the values for these three angles need to be learnt only once.
The sides of a 45°-45°-90° triangle are 1 : 1 : √2 and of a 30°-60°-90° triangle 1 : √3 : 2 (1 opposite 30°, √3 opposite 60°, hypotenuse 2). All values for 30°, 45°, 60° come from these.
In the figure: The sides of the two triangles are 1 : 1 : √2 and 1 : √3 : 2.
Rule 1: 30° and 60° — 30°: sin = 1/2, cos = √3/2, tan = 1/√3. 60°: sin = √3/2, cos = 1/2, tan = √3. (sin 30° is cos 60°.)
- Easy: sin 30° = 1/2 (opposite 1, hypotenuse 2).
- Medium: cos 60° = 1/2 and tan 60° = √3 ÷ 1 = √3.
- Exam type: sin 60° × cos 30° = (√3/2) × (√3/2) = 3/4.
Rule 2: 45° — 45°: sin = 1/√2, cos = 1/√2, tan = 1. It is an equal-leg triangle, so tan 45° = 1.
- Easy: tan 45° = 1.
- Medium: sin 45° × cos 45° = (1/√2) × (1/√2) = 1/2.
- Exam type: sin²45° + cos²45° = 1/2 + 1/2 = 1 (identity checked).
Rule 3: Adding and multiplying the values — Write the values as fractions. Easy order: brackets first, then multiply or divide, then add.
- Easy: sin 30° + cos 60° = 1/2 + 1/2 = 1.
- Medium: 2 × sin 30° × cos 30° = 2 × 1/2 × √3/2 = √3/2 ( = sin 60°).
- Exam type: tan 60° × tan 30° = √3 × 1/√3 = 1. And sin²30° + cos²30° = 1/4 + 3/4 = 1.
45°: tan = 1. For 30° and 60° the sin and cos values swap (1/2 and √3/2).
Writing sin 30° = √3/2. The side opposite 30° is the shortest (1), so sin 30° = 1/2.
Easy trick: for sin of 0°, 30°, 45°, 60°, 90° take √0, √1, √2, √3, √4 and divide by 2. Memory clue: sin: √0, √1, √2, √3, √4 each ÷ 2; cos in the reverse order.
Spot the difference
| Angle | sin | cos | tan |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
Practice questions
- 1What is sin 30°?
- 2What is tan 45°?
- 3What is cos 60°?
- 4Find sin 30° + cos 60°.
- 5What is tan 60° × tan 30°?
- 6What is sin 60° ÷ cos 60°?
- 7Find the value of sin²60° + cos²60°.
- 8Find the value of 2 sin 30° + tan 45° − cos 60°.
Show answers (try them yourself first)
- 11/2. 2) 1. 3) 1/2.
- 41/2 + 1/2 = 1. 5) √3 × 1/√3 = 1. 6) (√3/2) ÷ (1/2) = √3 (this is tan 60°).
- 73/4 + 1/4 = 1. 8) 2 × 1/2 + 1 − 1/2 = 1 + 1 − 1/2 = 1.5.
Heights and distances
Nobody climbs a tree to measure it. Stand at some distance on the ground, look at the top, and from the angle and distance the height can be found.
The angle made looking up from the horizontal line is the angle of elevation and the angle made looking down is the angle of depression. Both are measured from the horizontal line. They give a right triangle, and tan gives the height or distance: tan θ = height ÷ distance.
In the figure: Both angles start from the horizontal. Looking up is elevation, looking down is depression.
Heights and distances
In the app you can move this figure with your finger.
Open in the appRule 1: Use tan to find the height — Height = distance × tan θ. At 45° height = distance; at 60° height = distance × √3; at 30° height = distance ÷ √3.
- Easy: Distance 20 m, elevation 45° ⇒ height 20 × 1 = 20 m.
- Medium: Distance 30 m, angle 30° ⇒ height 30 ÷ √3 = 10√3 ≈ 17.3 m.
- Exam type: Distance 12 m, angle 60° ⇒ height 12√3 ≈ 20.8 m. The larger angle gives a greater height.
Rule 2: Find the distance — Distance = height ÷ tan θ. Given the height and the angle you get the distance from the point to the foot.
- Easy: Height 15 m, angle 45° ⇒ distance 15 m.
- Medium: Height 20 m, angle 60° ⇒ distance 20 ÷ √3 ≈ 11.5 m.
- Exam type: Height 9 m, angle 30° ⇒ distance 9 ÷ (1/√3) = 9√3 ≈ 15.6 m.
Rule 3: Angle of depression and two observations — The angle of depression equals the angle of elevation from the lower point (alternate angles). Find the distance to an object seen from a height as above. Do not forget to add the eye level.
- Easy: A tower 30 m high, a boat seen below at a depression of 45° ⇒ the boat is 30 m away.
- Medium: Tower 24 m, depression 60° ⇒ distance 24 ÷ √3 = 8√3 ≈ 13.9 m.
- Exam type: A boy's eye level is 1.5 m, the tree is 10 m away, angle 45° ⇒ tree height 10 + 1.5 = 11.5 m.
tan θ = height ÷ distance. At 45° height = distance. The angle is always from the horizontal line. Add the eye level at the end.
Measuring the angle from the vertical. Elevation and depression are always measured from the horizontal.
First draw the figure and mark the right triangle; the height is the opposite side and the distance is the adjacent side. Memory clue: Thirty is long, sixty is tall: at 30° the distance is bigger, at 60° the height is bigger.
Spot the difference
| Angle | tan | Height (distance d) |
|---|---|---|
| 30° | 1/√3 | d ÷ √3 |
| 45° | 1 | d |
| 60° | √3 | d × √3 |
Practice questions
- 1If the distance is 25 m and the angle of elevation 45°, what is the height?
- 2If the height is 40 m and the angle 45°, what is the distance?
- 3From which line is the angle of elevation measured?
- 4The distance is 15 m and the angle 60°. Find the height (√3 = 1.73).
- 5The height is 12 m and the angle 30°. Find the distance (√3 = 1.73).
- 6Tower 50 m, angle of depression 45°: how far is the object?
- 7A boy with eye level 1.5 m stands 18 m away and sees the top of a building at 30°. Find the height of the building (√3 = 1.73).
- 8A tree is 10 m high. What are the two angles of elevation from points 10 m and 10√3 m from its foot?
Show answers (try them yourself first)
- 125 × 1 = 25 m. 2) 40 m. 3) From the horizontal line.
- 415 × 1.73 = 25.95 m. 5) 12 ÷ (1/√3) = 12 × 1.73 = 20.76 m. 6) 50 m.
- 718 ÷ 1.73 ≈ 10.4, then add 1.5 ⇒ ≈ 11.9 m. 8) From 10 m: tan = 10/10 = 1 ⇒ 45°. From 10√3: tan = 10/(10√3) = 1/√3 ⇒ 30°.