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Online Trigonometry Tutoring: Trigonometric Identities

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Simplify trigonometry.

Our expert and qualified trigonometry tutors understand your problems and guide you towards better grades in trigonometry. Our aim is to help the student understand concepts in trigonometry and master the techniques of solving problems quickly with confidence.

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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