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

Sum & Difference Identities

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Sum & Difference Identities


The sum and difference identities, also known as Ptolemy's identities or compound formulas, are identities often used to evaluate non-special angles (i.e: π8\frac{\pi}{8}).

sin⁡(α+β)=sin⁡αcos⁡β+sin⁡βcos⁡αsin⁡(α−β)=sin⁡αcos⁡β−sin⁡βcos⁡αcos⁡(α+β)=cos⁡αcos⁡β−sin⁡αsin⁡βcos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡βtan⁡(α+β)=tan⁡α+tan⁡β1−tan⁡αtan⁡βtan⁡(α−β)=tan⁡α−tan⁡β1+tan⁡αtan⁡β\boxed{\begin{array}{rcl} \sin(\alpha+\beta)&=&\sin\alpha\cos\beta+\sin\beta\cos\alpha\\\\ \sin(\alpha-\beta)&=&\sin\alpha\cos\beta-\sin\beta\cos\alpha\\\\ \cos(\alpha+\beta)&=&\cos\alpha\cos\beta-\sin\alpha\sin\beta\\\\ \cos(\alpha-\beta)&=&\cos\alpha\cos\beta+\sin\alpha\sin\beta\\\\\\ \tan(\alpha+\beta)&=&\dfrac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}\\\\ \tan(\alpha-\beta)&=&\dfrac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta} \end{array}}


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Example

Let's evalute cos⁡(π12)\cos\Bigg(\dfrac{\pi}{12}\Bigg) using the sum and difference identities.

First, let's trying to express π12\dfrac{\pi}{12} as the sum or difference of two special angles.

π12=π3−π4\dfrac{\pi}{12}=\dfrac{\pi}{3}-\dfrac{\pi}{4}


Therefore, cos⁡(π12)=cos⁡(π3−π4)\cos\Bigg(\dfrac{\pi}{12}\Bigg)=\cos\Bigg(\dfrac{\pi}{3}-\dfrac{\pi}{4}\Bigg)


Then, using the difference identity for cos⁡θ\cos\theta:

cos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡βcos⁡(π3−π4)=cos⁡(π3)cos⁡(π4)+sin⁡(π3)sin⁡(π4)=(12)(12)+(32)(12)=1+322\begin{array}{rcl} \cos(\alpha-\beta)&=&\cos\alpha\cos\beta+\sin\alpha\sin\beta\\\\ \cos\Bigg(\dfrac{\pi}{3}-\dfrac{\pi}{4}\Bigg)&=&\cos\Bigg(\dfrac{\pi}{3}\Bigg)\cos\Bigg(\dfrac{\pi}{4}\Bigg)+\sin\Bigg(\dfrac{\pi}{3}\Bigg)\sin\Bigg(\dfrac{\pi}{4}\Bigg)\\\\ &=&\Bigg(\dfrac{1}{2}\Bigg)\Bigg(\dfrac{1}{\sqrt{2}}\Bigg)+\Bigg(\dfrac{\sqrt{3}}{2}\Bigg)\Bigg(\dfrac{1}{2}\Bigg)\\\\ &=&\boxed{\dfrac{1+\sqrt{3}}{2\sqrt{2}}} \end{array}
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Example: Sum & Difference Identities


Express 5π12\dfrac{5\pi}{12} as the sum or difference of two special angles.


Let 5π12=aπ12+bπ12\dfrac{5\pi}{12}=\dfrac{a\pi}{12}+\dfrac{b\pi}{12}, where a & ba~\&~b are integers that not relatively prime to 12 and that add to 5.


So, choose a=3a=3 and b=2b=2.

∴ 5π12=3π12+2π12=π4+π6\begin{array}{rcl} \therefore~\dfrac{5\pi}{12}&=&\dfrac{3\pi}{12}+\dfrac{2\pi}{12}\\\\ &=&\dfrac{\pi}{4}+\dfrac{\pi}{6} \end{array}

Practice: Sum & Difference Identities


Find two special angles whose difference is 7π12\dfrac{7\pi}{12}.

Practice: Sum & Difference Identities

Evaluate cos⁡(−π12)\cos\Bigg(-\dfrac{\pi}{12}\Bigg). Leave answer in exact form.

Practice: Sum & Difference Identities


If sin⁡(α+β)=sin⁡(23π12)\sin(\alpha+\beta)=\sin\Bigg(\dfrac{23\pi}{12}\Bigg), then what is sin⁡(α−β)?\sin(\alpha-\beta)?

Practice: Sum & Difference Identities

Using sum and difference identities, give the exact solution to tan⁡(31π12)\tan\Bigg(\dfrac{31\pi}{12}\Bigg).

Practice: Sum & Difference Identities


Determine the exact value for sin⁡(A+B)\sin(A+B) if cos⁡A=513\cos{A}=\dfrac{5}{13} and sin⁡B=45\sin{B}=\dfrac{4}{5}. Assume ∠A & ∠B\angle{A}~\&~\angle{B} are both in quadrant I.
Extra Practice
Sum and Difference Identities
Determine the exact value of tan⁡13π12\tan\dfrac{13\pi}{12}.
Sum and Difference Identities
Practice: Sum & Difference Identities
Determine the exact value for sin⁡(A−B)\sin(\text{A}-\text{B}) if cos⁡A=−19\cos{\text{A}}=-\dfrac{1}{9} and sin⁡B=411\sin{\text{B}}=\dfrac{4}{11}.
Sum and Difference Identities
Practice: Sum & Difference Identities
Determine the exact value for cos⁡(A+B)\cos(\text{A}+\text{B}) if cos⁡A=25\cos{\text{A}}=\dfrac{2}{5} and sin⁡B=37\sin{\text{B}}=\dfrac{3}{7}. Assume ∠A & ∠B\angle{\text{A}}~\&~\angle{\text{B}} are both in quadrant I.
Sum and Difference Identities
Practice: Sum & Difference Identities
Using sum and difference identities, give the exact solution to tan⁡(5π12)\tan\Bigg(\dfrac{5\pi}{12}\Bigg).
Sum and Difference Identities
Practice: Sum & Difference Identities
If cos⁡(α−β)=cos⁡(17π12)\cos(\alpha-\beta)=\cos\Bigg(\dfrac{17\pi}{12}\Bigg), then what is cos⁡(α+β)?\cos(\alpha+\beta)?
Sum and Difference Identities
Practice: Sum & Difference Identities
Evaluate sin⁡(11π12)\sin\Bigg(\dfrac{11\pi}{12}\Bigg). Leave answer in exact form.
Sum and Difference Identities
Practice: Sum & Difference Identities
Find two special angles whose difference is π12\frac{\pi}{12}.