Angle Addition Worksheet
Angle Addition Identities
Trigonometric Sum & Difference Formulas
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Core Sum & Difference Formulas Common Angles: \(30^\circ (\frac{\pi}{6})\), \(45^\circ (\frac{\pi}{4})\), \(60^\circ (\frac{\pi}{3})\), \(90^\circ (\frac{\pi}{2})\)
Sine Formulas
\(\sin(A + B) = \sin A \cos B + \cos A \sin B\)
\(\sin(A - B) = \sin A \cos B - \cos A \sin B\)
Cosine Formulas
\(\cos(A + B) = \cos A \cos B - \sin A \sin B\)
\(\cos(A - B) = \cos A \cos B + \sin A \sin B\)
Tangent Formulas
\(\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\)
\(\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}\)
Worked Example: Find the exact value of \(\sin(75^\circ)\)
Step 1: Decompose
Express as sum or difference of special angles:
\(75^\circ = 45^\circ + 30^\circ\)
Step 2: Apply Identity
Substitute into sine sum rule:
\(\sin(45^\circ)\cos(30^\circ) + \cos(45^\circ)\sin(30^\circ)\)
Step 3: Evaluate
Use special triangle values:
\((\frac{\sqrt{2}}{2})(\frac{\sqrt{3}}{2}) + (\frac{\sqrt{2}}{2})(\frac{1}{2})\)
Step 4: Simplify
Combine over common denominator:
\(\frac{\sqrt{6} + \sqrt{2}}{4}\)
Practice Exercises
Find the exact value of each. Show all steps and simplify fully.
1 Find exact value: \(\cos(75^\circ)\)
45° + 30°
Exact Value:
2 Find exact value: \(\sin(15^\circ)\)
45° - 30°
Exact Value:
3 Find exact value: \(\cos(105^\circ)\)
60° + 45°
Exact Value:
4 Find exact value: \(\sin(105^\circ)\)
60° + 45°
Exact Value:
5 Find exact value: \(\cos(15^\circ)\)
45° - 30°
Exact Value:
6 Find exact value: \(\tan(75^\circ)\)
45° + 30°
Exact Value:
Tip: Pay close attention to the sign change in the cosine identity: \(\cos(A+B)\) uses a minus sign, while \(\cos(A-B)\) uses a plus sign. Trig Identities · Unit 4
Angle Addition Answer Key
Angle Addition Identities
Teacher Answer Key
Complete Step-by-Step Solutions & Pedagogical Notes
Scoring Suggestion: 3 pts / problem
1 pt setup · 1 pt substitution · 1 pt exact answer
Common Student Misconceptions to Watch: \(\sqrt{6}\pm\sqrt{2}\) cannot simplify to \(\sqrt{8}\) or \(\sqrt{4}\)
Cosine Sign Trap: Students often write \(\cos(A+B)=\cos A\cos B + \sin A\sin B\). Emphasize the sign flips for cosine.
Quadrant Signs: Check QII angles like \(105^\circ\). Cosine must be negative: \(\frac{\sqrt{2}-\sqrt{6}}{4} < 0\).
Co-function Symmetry: Point out why \(\cos(75^\circ) = \sin(15^\circ)\) and \(\sin(105^\circ) = \cos(15^\circ) = \sin(75^\circ)\).
1 Evaluate \(\cos(75^\circ)\)
\(45^\circ + 30^\circ\)
Identity: \(\cos(A+B) = \cos A \cos B - \sin A \sin B\)
\(= \cos 45^\circ \cos 30^\circ - \sin 45^\circ \sin 30^\circ\)
\(= (\frac{\sqrt{2}}{2})(\frac{\sqrt{3}}{2}) - (\frac{\sqrt{2}}{2})(\frac{1}{2})\)
\(= \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}\)
Notice: \(\cos 75^\circ = \sin 15^\circ\)
\(\frac{\sqrt{6} - \sqrt{2}}{4}\)
2 Evaluate \(\sin(15^\circ)\)
\(45^\circ - 30^\circ\)
Identity: \(\sin(A-B) = \sin A \cos B - \cos A \sin B\)
\(= \sin 45^\circ \cos 30^\circ - \cos 45^\circ \sin 30^\circ\)
\(= (\frac{\sqrt{2}}{2})(\frac{\sqrt{3}}{2}) - (\frac{\sqrt{2}}{2})(\frac{1}{2})\)
\(= \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}\)
Also valid: \(60^\circ - 45^\circ\)
\(\frac{\sqrt{6} - \sqrt{2}}{4}\)
3 Evaluate \(\cos(105^\circ)\)
\(60^\circ + 45^\circ\)
Identity: \(\cos(A+B) = \cos A \cos B - \sin A \sin B\)
\(= \cos 60^\circ \cos 45^\circ - \sin 60^\circ \sin 45^\circ\)
\(= (\frac{1}{2})(\frac{\sqrt{2}}{2}) - (\frac{\sqrt{3}}{2})(\frac{\sqrt{2}}{2})\)
\(= \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4}\)
Quadrant II: Result is negative
\(\frac{\sqrt{2} - \sqrt{6}}{4}\)
4 Evaluate \(\sin(105^\circ)\)
\(60^\circ + 45^\circ\)
Identity: \(\sin(A+B) = \sin A \cos B + \cos A \sin B\)
\(= \sin 60^\circ \cos 45^\circ + \cos 60^\circ \sin 45^\circ\)
\(= (\frac{\sqrt{3}}{2})(\frac{\sqrt{2}}{2}) + (\frac{1}{2})(\frac{\sqrt{2}}{2})\)
\(= \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}\)
Equivalent to \(\sin(75^\circ)\)
\(\frac{\sqrt{6} + \sqrt{2}}{4}\)
5 Evaluate \(\cos(15^\circ)\)
\(45^\circ - 30^\circ\)
Identity: \(\cos(A-B) = \cos A \cos B + \sin A \sin B\)
\(= \cos 45^\circ \cos 30^\circ + \sin 45^\circ \sin 30^\circ\)
\(= (\frac{\sqrt{2}}{2})(\frac{\sqrt{3}}{2}) + (\frac{\sqrt{2}}{2})(\frac{1}{2})\)
\(= \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}\)
Notice sign flip in cosine diff
\(\frac{\sqrt{6} + \sqrt{2}}{4}\)
6 Evaluate \(\tan(75^\circ)\)
\(45^\circ + 30^\circ\)
Identity: \(\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\)
\(= \frac{\tan 45^\circ + \tan 30^\circ}{1 - \tan 45^\circ \tan 30^\circ} = \frac{1 + \frac{\sqrt{3}}{3}}{1 - \frac{\sqrt{3}}{3}} = \frac{3 + \sqrt{3}}{3 - \sqrt{3}}\)
Rationalize: \(\frac{(3+\sqrt{3})^2}{9-3} = \frac{12 + 6\sqrt{3}}{6}\)
Or via ratio: \(\frac{\sin 75^\circ}{\cos 75^\circ} = 2 + \sqrt{3}\)
Accept \(\frac{3+\sqrt{3}}{3-\sqrt{3}}\) or fully simplified
\(2 + \sqrt{3}\)
Trigonometric Angle Addition Identities · Teacher Solution Guide & Scoring Rubric Total Score: /18
