Sum and Difference of Angles Formulas

Sum and difference of angles formulas are used to find exact trig values for angles that can be written as the sum or difference of familiar unit-circle angles. These identities work for sine, cosine, and tangent, and they are especially helpful for angles like \(15^\circ\), \(75^\circ\), and \(195^\circ\). This page includes exact value problems, tangent sum and difference problems, and identity verification using angle formulas.

Notes

Notes for Sum and Difference Angle Formula

Practice Problems

Find the exact value.

\(\textbf{1)}\) \( \sin{\left(15°\right)} \)Link to Youtube Video Solving Question Number 1

 

\(\textbf{2)}\) \( \cos{\left(195°\right)} \)

 

\(\textbf{3)}\) \( \cos{\left(75°\right)} \)

 

\(\textbf{4)}\) \( \sin{\left(285°\right)} \)

 

\(\textbf{5)}\) \( \cos{\left(-15°\right)} \)

 

\(\textbf{6)}\) Simplify \( \displaystyle \frac{\tan{\frac{11\pi}{8}}-\tan{\frac{3\pi}{8}}}{1+\tan{\frac{11\pi}{8}}\tan{\frac{3\pi}{8}}} \)

 

\(\textbf{7)}\) \( \cos{\left(90°+\theta\right)}=-\sin{\theta} \)

 

\(\textbf{8)}\) \(\sin\left(75^\circ\right)\)

 

\(\textbf{9)}\) \(\cos\left(15^\circ\right)\)

 

\(\textbf{10)}\) \(\sin\left(105^\circ\right)\)

 

\(\textbf{11)}\) \(\tan\left(75^\circ\right)\)

 

\(\textbf{12)}\) \(\tan\left(15^\circ\right)\)

 

\(\textbf{13)}\) \(\sin\left(\frac{7\pi}{12}\right)\)

 

\(\textbf{14)}\) \(\cos\left(\frac{5\pi}{12}\right)\)

 

\(\textbf{15)}\) \(\tan\left(\frac{5\pi}{12}\right)\)

 

Challenge Problems

\(\textbf{16)}\) Verify \(\sin\left(\frac{\pi}{2}+x\right)=\cos{x}\)

 

\(\textbf{17)}\) Verify \(\cos\left(\frac{\pi}{2}-x\right)=\sin{x}\)

 

\(\textbf{18)}\) Simplify \(\sin(x+y)-\sin(x-y)\)

 

\(\textbf{19)}\) Simplify \(\cos(x-y)-\cos(x+y)\)

 

\(\textbf{20)}\) Verify \(\tan\left(\frac{\pi}{4}+x\right)=\frac{1+\tan{x}}{1-\tan{x}}\)

 

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\(\bullet\text{ Graphing Trig Functions- sin and cos}\)
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\(\bullet\text{ Solving Trigonometric Equations}\)
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