Trigonometry deals with the relationship between the angles and sides of the triangles. It is derived from ‘trigon’ and ‘metron’ (Greek words) that means the measurement of the sides of a triangle. An angle is the measurement of the rotation of a revolving line w.r.t to a fixed line. The angle has +ve or -ve values depending on its rotation [-ve for clockwise rotation and +ve for anti-clockwise rotation].
Conventions for Angle measurements
- Sexagesimal system: In this system, the angle is measured in degrees. 1° = 60′ and 1′ = 60″.
- Circular system: In this system, the angle is measured in radian.
1 radian = Angle subtended at the center of a circle by an arc of length equal to the radius of the circle.
Relationships between Radian and degree
1 Radian = 180/π degree and 1 Degree = π/180 radian
Functions of negative angles (A)
- sin (-A) = – sinA,
- cosec (-A) = -cosec A,
- tan (-A) = – tan A,
- cos (-A) = cos A,
- cot (-A) = -cot A,
- sec (-A) = sec A.
Trigonometric formulas of compound angles
The resultant of the sum or differences of two or more angles is known as the compound angle that results in trigonometric identities as mentioned below.
- sin (C + D) = + cosC sinD + sinC cosD
- sin (C – D) = sinC cosD – cosC sinD
- cos (C + D) = cosC cosD – sinC sinD
- cos (C – D) = sinC sinD + cos C cosD
- 2 sinC cosD = sin (C + D) + sin (C – D)
- 2 cosC sinD = sin (C + D) – sin (C – D)
- 2 cosC cosD = cos (C + D) + cos (C – D)
- 2 sinC sinD = cos (C – D) – cos (C + D)
Important Trigonometric equations
The equations involving trigonometric functions are known as trigonometric equations. These trigonometric equations are termed as identities if all the values of unknown angles for which the functions are defined are satisfied. The solutions of trigonometric expressions for which 0 ≤ θ < 2π is known as the principal solution. The equations involving integer ‘n’ which provides all the possible solutions of a trigonometric expression is known as the general solution.
Trigonometric Equations General Solution
If sin A = sin P, then A = nπ + (–1)nP for n ∈ Z gives the general solution of the given trigonometric equation. If cos A = cos P, then A = 2nπ ± P, n ∈ Z gives the general solution of the given trigonometric equation. If tan A = tan P or cot A = cot P, then A = nπ + P, n ∈ Z, gives general solution for both the given trigonometric equations.
The general value of A satisfying equations sin 2A = sin 2P, cos 2A = Cos 2P and tan 2A = tan 2P is given by A = nπ ± P. The general value of A satisfying equations sin A = sin P and cos A = cos P simultaneously is given by A = 2nπ + P, n ∈ Z.
|NCERT Solutions for Class 11 Maths Chapter 3|
|NCERT Exemplar for Class 11 Maths Chapter 3|
Trigonometric Functions Class 11 Practice Questions
- Determine the value of [-cos (45° + (-θ))] + sin (θ + 45°).
- Evaluate tan 3P + tan 2P – tan P
- Find the minimum value of 4cosx – 5sinx – 87.
- If tan θ = 3 and θ lies in the 4th quadrant, then find the value of sin θ.
- Find the value of sin1° sin2° sin3° . . . . . . . . . sin 179°.
- Find the general solution of trigonometric equation (-3)sin2x + sin3x = 3 cos2x + cosx – cos3x
- Determine the general value of θ, If cosθ + sinθ = 1.
- If 3tan (θ – 20°) = tan (θ + 20°), 0° < θ < 90°, then find the value of θ.
- Solve the equation sin 2θ + sin θ + sin 3θ = 0.
- Determine the value of tan 19° + tan 127° – tan 163° – tan 21°.