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Wize High School Grade 12 Pre-Calculus Textbook > Trigonometric Identities & Proofs

Equivalent Trigonometric Expressions

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Equivalent Trigonometric Expressions


The following are cofunction identities that describe complementary angles derived from the right triangle.

sin⁡θ=cos⁡(π2−θ)cos⁡θ=sin⁡(π2−θ)tan⁡θ=cot⁡(π2−θ)csc⁡θ=sec⁡(π2−θ)sec⁡θ=csc⁡(π2−θ)cot⁡θ=tan⁡(π2−θ)\begin{array}{rcl} \sin\theta&=&\cos\Bigg(\displaystyle\frac{\pi}{2}-\theta\Bigg)\\\\ \cos\theta&=&\sin\Bigg(\displaystyle\frac{\pi}{2}-\theta\Bigg)\\\\ \tan\theta&=&\cot\Bigg(\displaystyle\frac{\pi}{2}-\theta\Bigg)\\\\ \csc\theta&=&\sec\Bigg(\displaystyle\frac{\pi}{2}-\theta\Bigg)\\\\ \sec\theta&=&\csc\Bigg(\displaystyle\frac{\pi}{2}-\theta\Bigg)\\\\ \cot\theta&=&\tan\Bigg(\displaystyle\frac{\pi}{2}-\theta\Bigg) \end{array}


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Example

What is sin⁡(π6)\sin\Bigg(\displaystyle\frac{\pi}{6}\Bigg) equivalent to?

sin⁡(π6)=cos⁡(π2−π6)sin⁡(π6)=cos⁡(π3) \begin{array}{rcl} \sin\Bigg(\displaystyle\frac{\pi}{6}\Bigg)&=&\cos\Bigg(\displaystyle\frac{\pi}{2}-\displaystyle\frac{\pi}{6}\Bigg)\\\\ \sin\Bigg(\displaystyle\frac{\pi}{6}\Bigg)&=&\cos\Bigg(\displaystyle\frac{\pi}{3}\Bigg) \end{array}
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Example: Equivalent Trigonometric Expressions


If sin⁡(11π13)=cos⁡θ\sin\Bigg(\displaystyle\frac{11\pi}{13}\Bigg)=\cos\theta, then what is θ\theta?


From the cofunction identity,

sin⁡θ=cos⁡(π2−θ)sin⁡(11π13)=cos⁡(π2−11π13)sin⁡(11π13)=cos⁡(−9π26)\begin{array}{rcl} \sin\theta&=&\cos\Bigg(\displaystyle\frac{\pi}{2}-\theta\Bigg)\\\\ \sin\Bigg(\displaystyle\frac{11\pi}{13}\Bigg)&=&\cos\Bigg(\displaystyle\frac{\pi}{2}-\displaystyle\frac{11\pi}{13}\Bigg)\\\\ \sin\Bigg(\displaystyle\frac{11\pi}{13}\Bigg)&=&\cos\Bigg(\displaystyle-\frac{9\pi}{26}\Bigg) \end{array}


Therefore, θ=−9π26\boxed{\theta=-\displaystyle\frac{9\pi}{26}}.

Practice: Equivalent Trigonometric Expressions


Let tan⁡(3π7)=cot⁡θ\tan\Bigg(\displaystyle\frac{3\pi}7{\Bigg)}=\cot\theta. What is θ\theta?

Practice: Equivalent Trigonometric Expressions

Assume cot⁡(π3)\cot\Bigg(\displaystyle\frac{\pi}{3}\Bigg). Evaluate tan⁡θ\tan\theta using the cofunction identity.

Practice: Equivalent Trigonometric Expressions


Let cos⁡θ=a\cos\theta=a. What is sin⁡(θ−π2)\sin\Bigg(\theta-\displaystyle\frac{\pi}{2}\Bigg)?
Extra Practice
Equivalent Trigonometric Expressions
Practice: Equivalent Trigonometric Expressions
Let sec⁡θ=x+y\sec\theta=x+y. What is sin⁡(θ−π2)\sin\Bigg(\theta-\displaystyle\frac{\pi}{2}\Bigg)?
Equivalent Trigonometric Expressions
Practice: Equivalent Trigonometric Expressions
Let sin⁡θ=x+5\sin\theta=x+5. What is cos⁡(θ−π2)\cos\Bigg(\theta-\dfrac{\pi}{2}\Bigg)?
Equivalent Trigonometric Expressions
Practice: Equivalent Trigonometric Expressions
Assume cot⁡(π4)\cot\Bigg(\dfrac{\pi}{4}\Bigg). Evaluate tan⁡θ\tan\theta using the cofunction identity.
Equivalent Trigonometric Expressions
Practice: Equivalent Trigonometric Expressions
Assume cot⁡(5π6)\cot\Bigg(\dfrac{5\pi}{6}\Bigg). Evaluate tan⁡θ\tan\theta using the cofunction identity.
Equivalent Trigonometric Expressions
Practice: Equivalent Trigonometric Expressions
Let tan⁡(4π11)=cot⁡θ\tan\Bigg(\dfrac{4\pi}{11}{\Bigg)}=\cot\theta. What is θ\theta?
Equivalent Trigonometric Expressions
Practice: Equivalent Trigonometric Expressions
Let tan⁡(5π8)=cot⁡θ\tan\Bigg(\dfrac{5\pi}{8}{\Bigg)}=\cot\theta. What is θ\theta?