top of page
Search

Verifying Trigonometric Identities Step-by-Step (Example with θ = 240°)

  • Writer: Tyler Buffone
    Tyler Buffone
  • Oct 19
  • 1 min read
Abstract art with overlapping orange and blue circles and arcs on a dark blue background, creating a harmonious and balanced composition.

Verify the following equation at θ = 240° by substituting this angle and comparing both sides.


Mathematical equation on a dark background: Sinθ/Cscθ + Cosθ/Secθ = Sec²θ - Tan²θ, written in white handwriting.

Step 1: Rewrite any reciprocal trigonometric functions (secant, cosecant, and cotangent) in terms of their primary ratios (cosine, sine, and tangent, respectively). Make sure not to move anything over the equal sign from left to right or from right to left.


Remember that dividing by a fraction is the same as multiplying by its reciprocal.


For instance:


A / (B / C) = A x (C / B)


In the context of this question, we see this fact playing out as follows:


sinθ / (1 / sinθ) = sinθ x (sinθ / 1) = sinθ x sinθ = sin²θ

cosθ / (1 / cosθ) = cosθ x (cosθ / 1) = cosθ x cosθ = cos²θ


Mathematical equations featuring trigonometric identities, including Sinθ, Cosθ, Tanθ in white and yellow text, on a dark background.

Note that sin²θ + cos²θ = 1


Therefore, we obtain this simplified form of the identity, which will make it easier when we substitute the given value of θ.


Mathematical identity in white and yellow text on a dark background: 1 = 1/cos²θ - tan²θ.

Step 2: Let θ = 240°, evaluate the remaining trigonometric ratios, and then simplify. If your class permits you to use a calculator, then you may do so. You should end up seeing that both sides of the equation are the same, which verifies the identity.


Mathematical equation simplifying 1 equals 1 using trigonometric identities, with white and orange text on a dark background.


Need help with Verifying Trigonometric Identities? If you’re in Winnipeg and looking for a tutor, Tutor Advance provides expert one-on-one support!



 
 
bottom of page