Trigonometrical Ratios
ICSE Class 9 · Mathematics · Chapter 21
What this chapter covers
- What trigonometry measures
- Naming the sides of a right-angled triangle
- Sine, cosine and tangent of an acute angle
- The reciprocal ratios: cosecant, secant and cotangent
- Relations between the ratios
- Reading and writing trigonometrical notation
- Why the ratios depend only on the angle
- How large or small each ratio can be
- Finding the ratios when two sides are known
- Finding all the ratios when one ratio is known
- Evaluating expressions involving the ratios
- Common errors and how to avoid them
Key points
- Trigonometry studies the relationships between the angles and the sides of triangles; in Class IX every ratio is defined for an acute angle of a right-angled triangle.
- The hypotenuse is opposite the right angle; the opposite and adjacent sides depend on which acute angle is chosen.
- sin θ = (opp)/(hyp), cos θ = (adj)/(hyp), tan θ = (opp)/(adj).
- cosec θ = (1)/(sin θ), sec θ = (1)/(cos θ), cot θ = (1)/(tan θ).
- tan θ = (sin θ)/(cos θ) and cot θ = (cos θ)/(sin θ).
- Each ratio is a pure number and depends only on the angle, not on the size of the triangle.
- For an acute angle, sine and cosine lie between 0 and 1, cosecant and secant are greater than 1, and tangent and cotangent can be any positive number.
- Given one ratio, take the two sides as multiples of k, find the third side by Pythagoras' theorem, and write the rest.
- sin2 θ means (sin θ)2; sin θ is never "sin times θ".
Questions answered
Define the sine and the tangent of an acute angle in a right-angled triangle.
The sine of an acute angle is the ratio of the side opposite the angle to the hypotenuse. The tangent is the ratio of the side opposite the angle to the side adjacent to it.
In triangle LMN, ∠ M = 90°. Name the opposite side, the adjacent side and the hypotenuse for ∠ N.
Opposite side LM, adjacent side MN, hypotenuse LN.
Which ratio is the reciprocal of the cosine? Give its definition in terms of sides.
The secant: sec θ = (hypotenuse)/(adjacent).
In triangle ABC, ∠ B = 90°, AB = 15 cm and BC = 8 cm. Find sin A, cos A and tan A.
AC = √(225 + 64) = 17 cm. sin A = (8)/(17), cos A = (15)/(17), tan A = (8)/(15).
In the same triangle, find cosec C, sec C and cot C.
For ∠ C, opposite = AB = 15 cm and adjacent = BC = 8 cm. cosec C = (17)/(15), sec C = (17)/(8), cot C = (8)/(15).
If cos θ = (7)/(25), find the other five ratios of θ.
Opposite = √(625 - 49) k = 24k. sin θ = (24)/(25), tan θ = (24)/(7), cosec θ = (25)/(24), sec θ = (25)/(7), cot θ = (7)/(24).
If sin A = (1)/(√(2)), find cos A and tan A.
Hypotenuse √(2) k, opposite k, adjacent √(2k2 - k2) = k. So cos A = (1)/(√(2)) = (√(2))/(2) and tan A = 1.
Why can the sine of an acute angle never be equal to (9)/(8)?
The sine is opposite over hypotenuse, and the hypotenuse is the longest side of a right-angled triangle, so the sine is always less than 1; (9)/(8) is greater than 1.