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  1. Glossary
  2. /Mathematics
  3. /Trigonometric Ratios

What are Trigonometric Ratios?

In short

The trigonometric ratios sine, cosine and tangent compare the sides of a right-angled triangle: sin⁡θ=oppositehypotenuse\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}sinθ=hypotenuseopposite​, cos⁡θ=adjacenthypotenuse\cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}cosθ=hypotenuseadjacent​ and tan⁡θ=oppositeadjacent\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}tanθ=adjacentopposite​.

In simple words

The ratios depend only on the angle, not on the size of the triangle. tan⁡θ=sin⁡θcos⁡θ\tan\theta = \dfrac{\sin\theta}{\cos\theta}tanθ=cosθsinθ​, and sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1sin2θ+cos2θ=1. As an angle approaches 90∘90^\circ90∘, tan⁡θ\tan\thetatanθ becomes very large, because the adjacent side becomes very small: tan⁡89.9∘≈573\tan 89.9^\circ \approx 573tan89.9∘≈573. At exactly 90∘90^\circ90∘, tan⁡θ\tan\thetatanθ is not defined. Remember the values at 30∘30^\circ30∘, 45∘45^\circ45∘ and 60∘60^\circ60∘.

Example

A ladder 10 m10\ \text{m}10 m long makes 60∘60^\circ60∘ with the ground. It reaches a height of 10 m×sin⁡60∘=8.66 m10\ \text{m} \times \sin 60^\circ = 8.66\ \text{m}10 m×sin60∘=8.66 m up the wall.

Also searched as: tan 89.9, tan theta, sine cosine tangent.

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Related terms

  • Pythagoras Theorem
  • Radian

← All glossary terms

Learn it in

  1. Trigonometric RatiosClass 9 to 10
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Practise

  1. Find tan 89° to two decimal places with a calculator or Python…Medium
  2. A ramp rises at 30° to the ground and reaches a doorstep 0.9 m high…Medium
  3. A and B are angles with A > B and 0° < A + B ≤ 90°…Medium
  4. Find the value of 2 cos² 45° + 3 tan² 60° - 4 sin² 30°.
Easy
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Quick questions

  1. For an acute angle A, tan A = 20/99. Which set of sides fits a right triangle with this…Hard
  2. A flagpole 24 m tall casts a shadow 10 m long on level ground. What is the sine of the…Hard
  3. In triangle PQR the right angle is at Q, PQ = 36 cm and QR = 77 cm. What is sin P - cos R?Hard

More Mathematics terms

  1. Algebraic Identity
  2. Arithmetic Progression
  3. Complex Number
  4. Derivative
  5. Function
  6. Isosceles Triangle
All Mathematics terms →

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