Trigonometry

After completion of this chapter, students will be able to:

  • know about various trigonometric ratios, trigonometric identities and equations.
  • solve problems related to trigonometric ratios, trigonometric identities and equations.
  • solve problems of height and distance.

Notes

Trigonometric Ratios of Compound Angles

Let A and B two angles. Then their sum A + B or the difference A - B is called a compound angle. The sum or difference of any two or more than two angles is called a compound angle.

Trigonometric Ratios of Sub Multiple Angles

If A is an angle, then \(\frac{A}{2}\), \(\frac{A}{3}\), \(\frac{A}{4}\) etc. are called sub - multiple angles of A.

Transformation of Trigonometric Formulae

Transformation formulae Key to remember
2sinA cosB = sin(A + B) + sin(A - B) 2 sin. cos = sin + sin
2 cosA sinB = sin(A + B) - sin(A - B) 2 cos. sin = sin - sin
2 cosA cosB = cos(A + B) + cos(A - B) 2 cos. cos = cos + cos
2 sinnA sinB = cos (A - B) - cos(A + B)

2 sin. sin = cos - cos

sinC + sinD = 2sin(\(\frac{C + D}{2}\)) cos (\(\frac{C - D}{2}\)) sin + sin = 2sin. cos
sinC - sinD = 2 cos (\(\frac{C + D}{2}\)) sin (\(\frac{C - D}{2}\)) sin - sin = 2cos. sin
cosC + cosD = 2 cos (\(\frac{C + D}{2}\)) cos (\(\frac{C - D}{2}\)) cos + cos = 2cos. cos
cosC - cosD = -2 sin (\(\frac{C + D}{2}\)) sin (\(\frac{C - D}{2}\)) cos - cos = 2sin. sin

Conditional Trigonometric Identities

Conditional Trigonometric Identities.

Identities which are true under some given conditions are termed as conditional identities and 

in this section, we will deal some trigonometric identities which are bound to the condition of the sum of the angles of a triangle i.e. A + B + C =π

Solution of Trigonometric Equations

A method for finding angles.

(i) First of all, we determine the quadrant where the angle falls. For this, we use the all sin, tan, cos rule.

If sinθ is positive, the angleθ falls in the 1st and 2nd quadrants and if sinθ is negative, the angleθ falls in the 3rd and 4th quadrants. If cosθ is positive,θ lies in the 1st and 4th quadrants and if cosθ is negative,θ lies in the 2nd and 3rd quadrants. if tanθ is positive,θ lies in the first and third quadrants and if tanθ is negative,θ lies in second and fourth quadrants.

(ii) To find the angle in the first quadrant, we find the acute angle which satisfies the equation.

For example, if 2cosθ = 1

then cosθ = \(\frac{1}{2}\)

or, cosθ = cos600

So, θ = 600

(iii) To find the angle in the second quadrant, we subtract acute angleθ from 1800.

(iv) To find the angle in the third quadrant, we add acute angleθ to 1800

(v) To find the angle in the 4th quadrant, we subtract acute angleθ to 3600

(vi) To find the value ofθ from the equations like sinθ = 0, cosθ = 0, tanθ= 0, sinθ = 1, cosθ = 1, sinθ = -1, cosθ = -1. We should note the following results:

If sinθ = 0, thenθ = 00, 1800 or 3600 If sinθ = 1, thenθ = 900
If tanθ = 0, thenθ = 00,1800 or 3600 If sinθ = -1, thenθ = 2700
If cosθ = 0, thenθ = 900 or 2700 If cosθ = 1, thenθ = 00 or 3600

If cosθ = -1, then θ = 1800.

Heights and Distances

This note explains about the angle of depression and angle of elevation.