Online Trigonometry Tutoring: Trigonometric Identities

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Learn Trigonometric Identities from certified Online Trigonometry tutor:

Identity is an equation which is always true for all defined values

Triangle

Tangent and cotangent functions can be expressed in terms of the sine and cosine functions

tan x = \frac{sin x}{cos x}

cot x = \frac{cos x}{sin x}

These reciprocals are used to find the  value of trig functions:

sin x = \frac{1}{cosec x} ; cosec x = \frac{1}{sin x}

cos x = \frac{1}{sec x} ; sec x = \frac{1}{cos x}

tan x = \frac{1}{cot x} ; cot x = \frac{1}{tan x}

  • Pythagorean Identities is to help simplify trigonometric expressions.
  • Simplifying trigonometric expressions to create a factorable expression used to find missing trigonometric values.
  • Used to find missing trigonometric values

sin^2\theta + cos^2\theta = 1

tan^2\theta + 1 = sec^2\theta

1 + cot^2\theta = cosec^2\theta

Example 1

  • Prove that tan y/sin y = sec y

Expressing it in terms of sin y or cos y, we get

LHS = tan y × 1/sin y

= (sin y/cos y) × 1/sin y (tan A = sin A/cos A)

= 1/cos y

= sec y                 (since 1/cos A = sec A)

= RHS

Example 2

Verify the identities

1/sec2a = sin2a . cos2a + cos4a

RHS = cos2a (sin2a + cos2a)

= cos2a (1) {we know, sin2a + cos2a =1}

= cos2a

LHS = 1/sec2a  = 1/(1/ cos2a) = cos2a

Check Point

Problem 1. If sinθ = 2/3, find cosθ.

Problem 2. If cos θ = 5/11 and sin θ = 9/11, what is the value of tan θ?

Problem 3. Express  sin 67° + cos 65°  in terms of trigonometric ratios of angle  between 0 and 45 degrees.

Problem 4. If cos θ = .7, find the value of sec θ.

Answer Key

1. cos θ = \sqrt{5}/3
2. tan θ = 9/5
3. cos 23° + sin 25°
4. sec θ = 10/7

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