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Geometry

Right Triangle Trigonometry

Lesson

In a right triangle, three ratios of sides have special names: sine, cosine, and tangent. The mnemonic SOH-CAH-TOA tells you which side goes on top and which on bottom.

SOH-CAH-TOA

sin⁡θ=oppositehypotenuse\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}
cos⁡θ=adjacenthypotenuse\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}
tan⁡θ=oppositeadjacent\tan\theta = \frac{\text{opposite}}{\text{adjacent}}

The opposite side is across from the angle. The adjacent side is next to it (NOT the hypotenuse).

Worked example — 3-4-5 triangle

Legs 3 and 4, hypotenuse 5. For the angle opposite the side of length 3:

sin⁡=3/5,cos⁡=4/5,tan⁡=3/4\sin = 3/5, \quad \cos = 4/5, \quad \tan = 3/4

Common values to memorize

  • sin⁡30∘=0.5\sin 30^\circ = 0.5, cos⁡60∘=0.5\cos 60^\circ = 0.5
  • tan⁡45∘=1\tan 45^\circ = 1
  • sin⁡90∘=1\sin 90^\circ = 1, cos⁡0∘=1\cos 0^\circ = 1

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

3-4-5 triangle, sin⁡ of angle opposite 3\text{3-4-5 triangle, } \sin \text{ of angle opposite 3}

Problem 2

3-4-5, cos⁡ of angle opposite 3\text{3-4-5, } \cos \text{ of angle opposite 3}

Problem 3

3-4-5, tan⁡ of angle opposite 3\text{3-4-5, } \tan \text{ of angle opposite 3}

Problem 4

5-12-13, sin⁡ of angle opposite 5\text{5-12-13, } \sin \text{ of angle opposite 5}

Practice

Standard problems matching the lesson.

Problem 5

6-8-10, sin⁡ opp 6\text{6-8-10, } \sin \text{ opp 6}

Problem 6

6-8-10, cos⁡ opp 6\text{6-8-10, } \cos \text{ opp 6}

Problem 7

6-8-10, tan⁡ opp 6\text{6-8-10, } \tan \text{ opp 6}

Problem 8

5-12-13, cos⁡ opp 5\text{5-12-13, } \cos \text{ opp 5}

Problem 9

5-12-13, tan⁡ opp 5\text{5-12-13, } \tan \text{ opp 5}

Problem 10

8-15-17, sin⁡ opp 8\text{8-15-17, } \sin \text{ opp 8}

Problem 11

8-15-17, cos⁡ opp 8\text{8-15-17, } \cos \text{ opp 8}

Problem 12

8-15-17, tan⁡ opp 8\text{8-15-17, } \tan \text{ opp 8}

Problem 13

Right tri: opp=6, hyp=10. sin⁡θ?\text{Right tri: opp=6, hyp=10. } \sin\theta?

Problem 14

Right tri: adj=4, hyp=5. cos⁡θ?\text{Right tri: adj=4, hyp=5. } \cos\theta?

Problem 15

Right tri: opp=3, adj=4. tan⁡θ?\text{Right tri: opp=3, adj=4. } \tan\theta?

Problem 16

Ladder length 10 at 30∘. Height reached?\text{Ladder length 10 at } 30^\circ. \text{ Height reached?}

Problem 17

Ladder length 10 at 60∘. Distance from wall?\text{Ladder length 10 at } 60^\circ. \text{ Distance from wall?}

Problem 18

Right tri: opp=7, hyp=25. sin⁡θ?\text{Right tri: opp=7, hyp=25. } \sin\theta?

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

5-12-13, sin⁡ of angle opposite 12\text{5-12-13, } \sin \text{ of angle opposite 12}

Problem 20

5-12-13, cos⁡ of angle opposite 12\text{5-12-13, } \cos \text{ of angle opposite 12}

Problem 21

sin⁡30∘\sin 30^\circ

Problem 22

cos⁡60∘\cos 60^\circ

Problem 23

tan⁡45∘\tan 45^\circ

Problem 24

From 50 ft up, object 50 ft away. Angle of depression?\text{From 50 ft up, object 50 ft away. Angle of depression?}

Problem 25

Right triangle legs 9 and 12. Hypotenuse?\text{Right triangle legs 9 and 12. Hypotenuse?}

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