Trigonometry Notes

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Class 10 Trigonometry Notes

Class 10 Trigonometry Notes

Trigonometric Ratios

In a right-angled triangle (a triangle with one angle of 90 degrees), there are six trigonometric ratios based on the two acute angles: sine (sin), cosine (cos), tangent (tan), cosecant (cosec), secant (sec), and cotangent (cot). Consider a right-angled triangle ABC, where angle A is 90 degrees, and sides AB, BC, and AC are opposite, adjacent, and hypotenuse, respectively.

Trigonometric Ratios:

  • Sine (sin A): AB / AC
  • Cosine (cos A): BC / AC
  • Tangent (tan A): AB / BC
  • Cosecant (cosec A): 1 / sin A
  • Secant (sec A): 1 / cos A
  • Cotangent (cot A): 1 / tan A

Values of Trigonometric Ratios for Special Angles

For certain angles (0°, 30°, 45°, 60°, and 90°), the values of trigonometric ratios can be determined without using a calculator. These values are fundamental and are widely used in trigonometry.

  • sin 0° = 0, cos 0° = 1, tan 0° = 0, cosec 0° = ∞, sec 0° = 1, cot 0° = ∞
  • sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3, cosec 30° = 2, sec 30° = 2√3, cot 30° = √3
  • sin 45° = √2/2, cos 45° = √2/2, tan 45° = 1, cosec 45° = √2, sec 45° = √2, cot 45° = 1
  • sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3, cosec 60° = 2/√3, sec 60° = 2, cot 60° = 1/√3
  • sin 90° = 1, cos 90° = 0, tan 90° = ∞, cosec 90° = 1, sec 90° = ∞, cot 90° = 0

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables within their domains. These identities play a vital role in simplifying trigonometric expressions and solving trigonometric equations. Here are some of the fundamental trigonometric identities:

  • Pythagorean Identities:
    • sin^2(A) + cos^2(A) = 1
    • 1 + tan^2(A) = sec^2(A)
    • cot^2(A) + 1 = cosec^2(A)
  • Reciprocal Identities:
    • sec(A) = 1 / cos(A)
    • cosec(A) = 1 / sin(A)
    • cot(A) = 1 / tan(A)
  • Quotient Identities:
    • tan(A) = sin(A) / cos(A)
    • cot(A) = cos(A) / sin(A)
  • Co-Function Identities:
    • sin(90° - A) = cos(A)
    • cos(90° - A) = sin(A)
    • tan(90° - A) = cot(A)
    • cot(90° - A) = tan(A)
    • sec(90° - A) = cosec(A)
    • cosec(90° - A) = sec(A)
  • Even-Odd Identities:
    • sin(-A) = -sin(A)
    • cos(-A) = cos(A)
    • tan(-A) = -tan(A)
    • cot(-A) = -cot(A)
  • Double Angle Identities:
    • sin(2A) = 2 * sin(A) * cos(A)
    • cos(2A) = cos^2(A) - sin^2(A) = 2 * cos^2(A) - 1 = 1 - 2 * sin^2(A)
    • tan(2A) = (2 * tan(A)) / (1 - tan^2(A))
  • Half Angle Identities:
    • sin(A/2) = ±√[(1 - cos(A)) / 2]
    • cos(A/2) = ±√[(1 + cos(A)) / 2]
    • tan(A/2) = ±√[(1 - cos(A)) / (1 + cos(A))]
  • Sum and Difference Identities:
    • sin(A + B) = sin(A) * cos(B) + cos(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))
  • Product-to-Sum Identities:
    • sin(A) * sin(B) = (1/2) * [cos(A - B) - cos(A + B)]
    • cos(A) * cos(B) = (1/2) * [cos(A - B) + cos(A + B)]
    • sin(A) * cos(B) = (1/2) * [sin(A + B) + sin(A - B)]
  • Sum-to-Product Identities:
    • sin(A) + sin(B) = 2 * sin[(A + B)/2] * cos[(A - B)/2]
    • sin(A) - sin(B) = 2 * cos[(A + B)/2] * sin[(A - B)/2]
    • cos(A) + cos(B) = 2 * cos[(A + B)/2] * cos[(A - B)/2]
    • cos(A) - cos(B) = -2 * sin[(A + B)/2] * sin[(A - B)/2]

Example Problems:

Let's solve some example problems using trigonometric ratios and identities:

Example 1:

Consider a right-angled triangle with one acute angle measuring 30 degrees and the hypotenuse measuring 10 units. Find the length of the side opposite to the 30-degree angle and the side adjacent to it.

Solution:

Given: Angle A = 30°, Hypotenuse (AC) = 10 units

Using the trigonometric ratio for sine (sin A), we have:
sin 30° = (Length of side opposite to angle A) / (Hypotenuse)
1/2 = (Length of side opposite to 30°) / 10
Length of side opposite to 30° = 10 × (1/2) = 5 units

Using the trigonometric ratio for cosine (cos A), we have:
cos 30° = (Length of side adjacent to angle A) / (Hypotenuse)
√3/2 = (Length of side adjacent to 30°) / 10
Length of side adjacent to 30° = 10 × (√3/2) ≈ 8.66 units

Therefore, the length of the side opposite to the 30-degree angle is 5 units, and the length of the side adjacent to the 30-degree angle is approximately 8.66 units.

Example 2:

A building casts a shadow 24 meters long when the angle of elevation of the sun is 30 degrees. Determine the height of the building.

Solution:

Let the height of the building be h meters.

Given: Angle of elevation (A) = 30°, Length of shadow (AC) = 24 meters

Using the trigonometric ratio for tangent (tan A), we have:
tan 30° = (Length of side opposite to angle A) / (Length of side adjacent to angle A)
1/√3 = h / 24
h = 24 × (1/√3) ≈ 13.86 meters

Therefore, the height of the building is approximately 13.86 meters.