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Trigonometric Identities

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What Are Mathematical Identities?

A mathematical statement that is true regardless of what value is substituted for the variable is called an identity.

 

Mathematical Identities Continued

Is \(\csc\left(\theta\right)=\dfrac{1}{\sin\left(\theta\right)}\) an identity statement?

Is \(\cos\left(\theta\right)=\sin\left(\theta\right)\) an identity statement?

 

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Lesson Goals

  • Develop and identify the Pythagorean, quotient, and reciprocal identities.
  • Identify and apply strategies to prove trigonometric identities.

Try This 1

Prove that \(1=\sin\left(\theta\right)\cot\left(\theta\right)\sec\left(\theta\right)\).

Try This 2 — For Extension: Visualizing Six Trig Ratios

The point \(P\left(\cos\left(\theta\right),\sin\left(\theta\right)\right)\) is on the terminal arm of the acute angle \(\theta\) in standard position and the point \(O\) is the origin. The line perpendicular to \(OP\) at \(P\) has an \(x\)-intercept at \(A\) and a \(y\)-intercept at \(B\).

  1. Explain why \(\triangle OCP \sim \triangle BPO\).
  2. Determine expressions for the lengths \(BP\) and \(BO\) in terms of \(\theta\).
  3. Explain why \(\triangle OCP \sim \triangle OPA\).
  4. Determine expressions for the lengths \(PA\) and \(OA\) in terms of \(\theta\).

Point P is on the unit circle. Angle BPO is 90 degrees. A right angle is made at C by drawing a vertical line from P to the x-axis.