11th Class Trigonometric Functions

Trigonometric Functions

• Trigonometric functions are used to model real-life situations, such as tides, annual temperatures, and phases of the Moon.
• These functions are periodic, meaning they repeat regularly.

Angles of Any Magnitude

• Angles can be measured in any size, not just acute and obtuse angles.
• Angles can be measured in different quadrants:
  • 1st quadrant: 0° to 90°, all ratios are positive
  • 2nd quadrant: 90° to 180°, sin is positive, cos and tan are negative
  • 3rd quadrant: 180° to 270°, tan is positive, sin and cos are negative
  • 4th quadrant: 270° to 360°, cos is positive, sin and tan are negative
• The ASTC rule (All Stations To Central) helps to remember the signs of the ratios in each quadrant.

Trigonometric Ratios

• Trigonometric ratios can be found for angles greater than 360° by turning around the circle more than once.
• Negative angles can be measured in the opposite direction from positive angles.
• The ASTC rule also works for negative angles.

Trigonometric Identities

• Reciprocal trigonometric ratios are the reciprocals of the sine, cosine, and tangent ratios.
• Examples of reciprocal trigonometric ratios:
  • sin-1, cos-1, and tan-1
  • cosecant, secant, and cotangent (which are the reciprocals of sine, cosine, and tangent respectively)

Terminology

• Amplitude: the height from the center of a periodic function to the maximum or minimum values.
• Centre: the mean value of a periodic function that is equidistant from the maximum and minimum values.
• Period: the length of one cycle of a periodic function.
• Phase: a horizontal shift (translation).
• Identity: an equation that shows the equivalence of two algebraic expressions for all values of the variables.
• Inverse trigonometric functions: the sin-1, cos-1, and tan-1 functions, which are the inverse functions of the sine, cosine, and tangent functions respectively.### Reciprocal Trigonometric Ratios
• The reciprocal trigonometric ratios are: cosecant (cosec), secant (sec), and cotangent (cot)
• The formulas for these ratios are:
  • cosec θ = 1 / sin θ
  • sec θ = 1 / cos θ
  • cot θ = 1 / tan θ
• These ratios have the same signs as their related ratios in the different quadrants

Complementary Angles

• If ∠B = θ, then ∠A = 90° - θ
• Angles ∠B and ∠A are complementary because they add up to 90°
• The trigonometric ratios of complementary angles are equal:
  • sin θ = cos (90° - θ)
  • cos θ = sin (90° - θ)
  • tan θ = cot (90° - θ)
  • sec θ = cosec (90° - θ)
  • cot θ = tan (90° - θ)

Trigonometric Identities

• The tangent identity: tan θ = sin θ / cos θ
• The Pythagorean identities:
  • cos² θ + sin² θ = 1
  • 1 + tan² θ = sec² θ
  • 1 + cot² θ = cosec² θ
• These identities can be rearranged to give:
  • cos² θ = 1 - sin² θ
  • sin² θ = 1 - cos² θ

Sums and Differences of Angles

• The formulas for the trigonometric ratios of sums and differences of angles are:
  • cos (A - B) = cos A cos B + sin A sin B
  • cos (A + B) = cos A cos B - sin A sin B
  • sin (A + B) = sin A cos B + cos A sin B
  • sin (A - B) = sin A cos B - cos A sin B### Sum and Difference Identities
• The sum and difference identities for trigonometric functions are:
  • sin(A + B) = sin A cos B + cos A sin B
  • sin(A - B) = sin A cos B - cos A sin B
  • cos(A + B) = cos A cos B - sin A sin B
  • cos(A - B) = cos A cos B + sin A sin B
  • tan(A + B) = (tan A + tan B) / (1 - tan A tan B)
  • tan(A - B) = (tan A - tan B) / (1 + tan A tan B)

Double Angle Identities

• The double angle identities for trigonometric functions are:
  • sin 2A = 2 sin A cos A
  • cos 2A = cos^2 A - sin^2 A
  • tan 2A = 2 tan A / (1 - tan^2 A)

Product to Sum Identities

• The product to sum identities for trigonometric functions are:
  • cos A cos B = [cos(A + B) + cos(A - B)] / 2
  • sin A sin B = [cos(A - B) - cos(A + B)] / 2
  • sin A cos B = [sin(A + B) + sin(A - B)] / 2
  • cos A sin B = [sin(A + B) - sin(A - B)] / 2

The t-Formulas

• The t-formulas express sin A, cos A, and tan A in terms of t = tan(A/2):
  • sin A = 2t / (1 + t^2)
  • cos A = (1 - t^2) / (1 + t^2)
  • tan A = 2t / (1 - t^2)

Examples and Exercises

• Various examples and exercises are provided to practice and apply the trigonometric identities.