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TrigonometryTrigonometryHigh school

Right-Triangle Trigonometry

Sine, cosine and tangent as side ratios.

In a right triangle, each acute angle defines three ratios of sides. Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse and tangent is opposite over adjacent, often remembered as SOH-CAH-TOA. Because similar triangles have equal ratios, these values depend only on the angle.

With one side and one acute angle you can find every other side, and with two sides you can find the angles using inverse functions such as θ=tan⁡−1(oppadj)\theta = \tan^{-1}\left(\frac{\text{opp}}{\text{adj}}\right). This is how surveyors, engineers and navigators measure distances they cannot reach directly.

Key terms

Trig ratios
Opposite over hypotenuse, adjacent over hypotenuse, opposite over adjacent.
sin⁡θ=opphyp,  cos⁡θ=adjhyp,  tan⁡θ=oppadj\sin\theta = \frac{\text{opp}}{\text{hyp}},\; \cos\theta = \frac{\text{adj}}{\text{hyp}},\; \tan\theta = \frac{\text{opp}}{\text{adj}}

Key ideas

  • Label sides relative to the angle you are using; opposite and adjacent swap for the other acute angle.
  • Angle of elevation is measured up from the horizontal; angle of depression down from it.
  • Special triangles: 30∘30^\circ-60∘60^\circ-90∘90^\circ sides 1:3:21 : \sqrt{3} : 2; 45∘45^\circ-45∘45^\circ-90∘90^\circ sides 1:1:21 : 1 : \sqrt{2}.
  • Check the calculator is in degree mode for degree problems.

Worked example

From 40 m away, the angle of elevation to the top of a tower is 35∘35^\circ. How tall is the tower?

  1. Known: adjacent =40= 40; want opposite.
  2. tan⁡35∘=h40\tan 35^\circ = \frac{h}{40}.
  3. h=40tan⁡35∘≈40×0.7002h = 40 \tan 35^\circ \approx 40 \times 0.7002.
  4. h≈28.0h \approx 28.0 m.

Answer: about 28.028.0 m

Common mistakes

  • Using sine when the known sides are opposite and adjacent.
  • Calculator in radian mode for degree angles.
  • Using the hypotenuse as the adjacent side.

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

  1. 1.In a right triangle the hypotenuse is 10 and an angle is 30∘30^\circ. Find the opposite side.

    Show answer

    55

  2. 2.Find cos⁡60∘\cos 60^\circ.

    Show answer

    12\frac{1}{2}

  3. 3.Opposite 7, adjacent 7. Find the angle in degrees.

    Show answer

    45∘45^\circ

Keep practising

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