Trigonometric Functions Solved Examples (Class 11 Mathematics)
Trigonometric functions relate angles to ratios of sides in triangles and describe periodic phenomena. These examples cover conversions between angle measu
TL;DR: Trigonometric functions relate angles to ratios of sides in triangles and describe periodic phenomena. These examples cover conversions between angle…
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Trigonometric functions relate angles to ratios of sides in triangles and describe periodic phenomena. These examples cover conversions between angle measu
Trigonometric Functions — Solved Numerical Examples (Step by Step)
Example 1: Convert 225 degrees to radians.
Solution: To convert degrees to radians, multiply by π/180:
Radians = degrees × (π/180)
= 225 × (π/180)
= 225π/180
= 5π/4 radians
As a decimal: 5π/4 ≈ 3.927 radians
Example 2: Find the exact value of sin(5π/6).
Solution: 5π/6 = π - π/6
This angle is in the second quadrant.
Using the identity sin(π - θ) = sin(θ):
sin(5π/6) = sin(π - π/6) = sin(π/6) = 1/2
Alternatively, 5π/6 = 150°, which is 30° into the second quadrant where sine is positive and equals sin(30°) = 1/2
Example 3: Solve tan(x) = √3 for x in [0, 2π).
Solution: tan(x) = √3
The tangent function equals √3 at specific angles:
tan(π/3) = √3 and tan(π/3 + π) = √3
In the interval [0, 2π):
x = π/3 (where tan is positive, first quadrant)
x = π/3 + π = 4π/3 (where tan is positive, third quadrant)
In degrees: x = 60° or x = 240°
Example 4: Prove the identity: sin²(x) + cos²(x) = 1.
Solution: Consider a right triangle with opposite side = a, adjacent side = b, and hypotenuse = c.
sin(x) = a/c and cos(x) = b/c
sin²(x) + cos²(x) = (a/c)² + (b/c²)
= a²/c² + b²/c²
= (a² + b²) / c²
By the Pythagorean theorem, a² + b² = c²
Therefore: sin²(x) + cos²(x) = c²/c² = 1 ✓
Example 5: Find the period of the function f(x) = 3sin(2x - π/3).
Solution: For a function of the form f(x) = A sin(Bx + C), the period is 2π / |B|.
In this function: A = 3, B = 2, C = -π/3
Period = 2π / |2| = 2π/2 = π
Example 6: Simplify: cos(90° - θ).
Solution: Using the cofunction identity: cos(90° - θ) = sin(θ)
Or in radians: cos(π/2 - θ) = sin(θ)
This is because 90° - θ is the complement of θ, and cosine of an angle equals sine of its complement.
Tips
- Always remember the cofunction identities: sin(90° - θ) = cos(θ) and cos(90° - θ) = sin(θ).
- Use the unit circle to visualize angles and their trigonometric values, especially for common angles like 0°, 30°, 45°, 60°, 90°.
- When solving trigonometric equations, remember to find all solutions in the given interval, not just the principal value.
- The phase shift of a function f(x) = sin(Bx + C) or f(x) = cos(Bx + C) is -C/B.
Frequently Asked Questions
Why is radian measure used more often than degrees in higher mathematics?
Radians are more natural in calculus because the derivatives of trigonometric functions have simpler forms when angles are in radians. For example, d/dx[sin(x)] = cos(x) only when x is in radians. Degrees introduce unnecessary constants in derivatives.
What is the difference between sin⁻¹(x) and 1/sin(x)?
sin⁻¹(x) denotes the inverse sine function (arcsine), which returns the angle whose sine is x. The expression 1/sin(x) is the reciprocal of sine, also called cosecant (csc x). These are completely different: sin⁻¹(x) is a function that undoes sine, while 1/sin(x) is just a reciprocal.
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