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5.3: Introduction to Trigonometric Identities

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    203440

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    Prerequisite Skills
    • Simplifying Expressions
      • Multiplying and Dividing Rational Expressions
      • Adding and Subtracting Rational Expressions
      • Simplifying Compound Rational Expressions

    What is an Identity?

    Definition: Identity

    An equation that is true for all allowable variable values is called an identity.

    Ratio Identities

    Theorem: Ratio Identities

    For any angle not coterminal with \(90^{\circ}\) or \(270^{\circ}\),\[\tan \left(\theta\right)=\dfrac{\sin \left(\theta\right)}{\cos \left(\theta\right)}.\nonumber \]For any angle not coterminal with \(0^{\circ}\) or \(180^{\circ}\),\[\cot \left(\theta\right)=\dfrac{\cos \left(\theta\right)}{\sin \left(\theta\right)}\nonumber \]

    Example \( \PageIndex{ 1 } \)

    If \( \sin\left( \theta \right) = -\frac{1}{5} \) and \( \cos\left( \theta \right) = -\frac{2\sqrt{6}}{5} \), find the values of \( \tan\left( \theta \right) \) and \( \cot\left( \theta \right) \).

    Reciprocal Identities

    Theorem: Reciprocal Identities

    \[ \begin{array}{rclclcrclcl}
    \sin\left( \theta \right) & = & \dfrac{1}{\csc\left( \theta \right)} & \quad & (\text{provided } \csc\left( \theta \right) \neq 0) & \qquad & \csc\left( \theta \right) & = & \dfrac{1}{\sin\left( \theta \right)} & \quad & (\text{provided } \sin\left( \theta \right) \neq 0) \\[6pt] \cos\left( \theta \right) & = & \dfrac{1}{\sec\left( \theta \right)} & \quad & (\text{provided } \sec\left( \theta \right) \neq 0) & \qquad & \sec\left( \theta \right) & = & \dfrac{1}{\cos\left( \theta \right)} & \quad & (\text{provided } \cos\left( \theta \right) \neq 0) \\[6pt] \tan\left( \theta \right) & = & \dfrac{1}{\cot\left( \theta \right)} & \quad & (\text{provided } \cot\left( \theta \right) \neq 0) & \qquad & \cot\left( \theta \right) & = & \dfrac{1}{\tan\left( \theta \right)} & \quad & (\text{provided } \tan\left( \theta \right) \neq 0) \\[6pt] \end{array} \nonumber \]

    Example \(\PageIndex{2}\)

    If \( \cot\left( \theta \right) = \frac{8}{5} \), find \( \tan\left( \theta \right) \).

    Pythagorean Identities

    Theorem: The Pythagorean Identity

    For any angle \(\theta\),\[\cos^2 \left(\theta\right)+\sin^2 \left(\theta\right)=1.\nonumber \]

    Proof
    From the coordinate definition of the trigonometric functions, we know that \( r = \sqrt{x^2 + y^2} \). Thus,\[\begin{array}{rrclcr}
    & r & = & \sqrt{x^2 + y^2} & \quad & \\[6pt] \implies & r^2 & = & x^2 + y^2 & \quad & (\text{squaring both sides}) \\[6pt] \implies & 1 & = & \dfrac{x^2}{r^2} + \dfrac{y^2}{r^2} & \quad & (\text{dividing both sides by }r^2) \\[6pt] \implies & 1 & = & \left( \dfrac{x}{r} \right)^2 + \left( \dfrac{y}{r} \right)^2 & \quad & (\text{Laws of Exponents}) \\[6pt] \implies & 1 & = & \left( \cos\left( \theta \right) \right)^2 + \left( \sin\left( \theta \right) \right)^2 & \quad & (\text{definitions of the cosine and sine functions}) \\[6pt] \implies & 1 & = & \cos^2\left( \theta \right) + \sin^2\left( \theta \right) & \quad & (\text{using a better notation}) \\[6pt] \end{array} \nonumber \]
    Example \( \PageIndex{ 3} \)

    Evaluate \(5\left[\cos ^2\left(12^{\circ}\right)+\sin ^2\left(12^{\circ}\right)\right] \) without using a calculator.

    Corollary: Alternate Forms of the Pythagorean Identity

    \[\begin{array}{rclcrcl}
    \sin ^2 \left(\theta\right) & = & 1-\cos ^2 \left(\theta\right) & \implies & \sin\left( \theta \right) & = & \pm \sqrt{1 - \cos^2\left( \theta \right)} \\[6pt] \cos ^2 \left(\theta\right) & = & 1-\sin ^2 \left(\theta\right) & \implies & \cos\left( \theta \right) & = & \pm \sqrt{1 - \sin^2\left( \theta \right)} \\[6pt] \end{array} \nonumber \]

    Example \( \PageIndex{ 4} \)
    1. If \(\cos \left(\theta\right)=\frac{5}{13}\) and \(-90^{\circ}<\theta<0^{\circ}\), find \(\sin \left(\theta\right)\) and \(\tan \left(\theta\right)\).
    2. Find \(\sin \left(\theta\right)\) and \(\tan \left(\theta\right)\) by using an appropriate sketch.
    Theorem: Pythagorean Identities

    For any angle \(\theta\) where the functions are defined,\[\begin{array}{rcl}
    \cos^2 \left(\theta\right)+\sin^2 \left(\theta\right) & = & 1 \\[6pt] 1 +\tan^2 \left(\theta\right) & = & \sec^2\left( \theta \right) \\[6pt] \cot^2 \left(\theta\right)+1 & = & \csc^2\left( \theta \right) \\[6pt] \end{array} \nonumber \]

    This page titled 5.3: Introduction to Trigonometric Identities is shared under a CC BY-SA 12 license and was authored, remixed, and/or curated by Roy Simpson.