Trigonometric ratios and using angles to find heights
Trigonometry studies the relationship between the angles and sides of a right-angled triangle. For an acute angle θ, sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse and tan θ = opposite ÷ adjacent, and their reciprocals are cosec θ, sec θ and cot θ. Because the ratios depend only on the angle and not on the size of the triangle, a small table of standard values is enough: sin 30° = ½, cos 60° = ½, tan 45° = 1, sin 60° = √3/2 and sin 90° = 1.
The identity sin²θ + cos²θ = 1 comes from Pythagoras' theorem, and dividing it by cos²θ or by sin²θ gives 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ. These identities are used to simplify expressions and prove results. In heights and distances, the angle of elevation is measured upwards from the horizontal to an object above the observer, and the angle of depression is measured downwards to an object below. If the top of a tower is seen at an elevation of 60° from a point 20 m from its base, the tower's height is 20 × tan 60° = 20√3 m.
Why it matters
Trigonometry lets you find heights and distances that cannot be measured directly, such as the height of a hill or the width of a river. It is used in surveying, navigation, architecture, astronomy and engineering. In Class 11 the ratios are extended to angles of any size and become trigonometric functions, which physics needs for waves, forces and circular motion.
Key terms
- Sine ratio
- The ratio of the side opposite an angle to the hypotenuse in a right-angled triangle.
- Angle of elevation
- The angle between the horizontal and the line of sight to an object above the observer.
- Angle of depression
- The angle between the horizontal and the line of sight to an object below the observer.
- Trigonometric identity
- An equation involving trigonometric ratios that is true for every allowed angle.