Overview
This lecture covers the trigonometric addition and double angle formulas, how to apply them to manipulate and solve trigonometric expressions and equations, and the process of rewriting combinations of sine and cosine in the form ( r \sin(x + \alpha) ) or ( r \cos(x - \alpha) ).
Trigonometric Addition & Double Angle Formulas
- Sine addition: ( \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B )
- Cosine addition: ( \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B )
- Tangent addition: ( \tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B} )
- Double angle formulas:
( \sin 2A = 2\sin A \cos A )
( \cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A )
( \tan 2A = \frac{2\tan A}{1 - \tan^2 A} )
Manipulating Trigonometric Expressions
- Any expression of the form ( a\cos\theta + b\sin\theta ) can be written as ( r\cos(\theta \pm \alpha) ) or ( r\sin(\theta + \alpha) ).
- To convert, set ( a = r\cos\alpha ), ( b = r\sin\alpha ), and ( r = \sqrt{a^2 + b^2} ).
- ( \tan \alpha = \frac{b}{a} ), so ( \alpha = \arctan\left(\frac{b}{a}\right) ).
Exact Values of Standard Angles
- Sine: 0° (0), 30° (1/2), 45° ((\sqrt{2}/2)), 60° ((\sqrt{3}/2)), 90° (1).
- Cosine: 0° (1), 30° ((\sqrt{3}/2)), 45° ((\sqrt{2}/2)), 60° (1/2), 90° (0).
- Tangent: 0° (0), 30° ((\sqrt{3}/3)), 45° (1), 60° ((\sqrt{3})), 90° (undefined).
Solving Equations Using Addition and Double Angle
- Use identities to rewrite complex trigonometric expressions into solvable forms (e.g. quadratics in (\cos x) or (\sin x)).
- Always check allowed intervals for the angle; use the CAST diagram for sign and interval reasoning.
Worked Examples & Applications
- Given values for (\sin a) and (\cos b), use Pythagoras ((\sin^2 + \cos^2 = 1)) to find missing sides and compute combined angles using addition formulas.
- Express trigonometric equations as single terms using identities, then solve for angles as required, considering intervals.
- For maximum/minimum values of ( r\sin(\theta + \alpha) ) or ( r\cos(\theta - \alpha) ), the extrema are at ( r ) and occur when the inside angle equals 90° (for sine) or 0° (for cosine).
Proving Trigonometric Identities
- Substitute addition/double angle formulas as appropriate.
- Simplify stepwise, showing algebraic manipulation, matching both sides of the identity.
- Use relationships such as ( \cot x = 1/\tan x ) when helpful.
Key Terms & Definitions
- Addition Formula — Formulas to expand (\sin(A \pm B)), (\cos(A \pm B)), (\tan(A \pm B)) into functions of A and B.
- Double Angle Formula — Formulas for (\sin 2A), (\cos 2A), (\tan 2A) in terms of A.
- CAST Diagram — Tool for determining the sign of trig ratios in each quadrant.
- Pythagorean Identity — ( \sin^2 x + \cos^2 x = 1 ).
- r-Form — Expression of ( a\cos\theta + b\sin\theta ) as ( r\cos(\theta \pm \alpha) ) or ( r\sin(\theta + \alpha) ).
Action Items / Next Steps
- Complete textbook exercises: 4A (Q2, 5, 7, 9 a/c/d), 4B (Q3, 5, 6b/c, 9 a/b/c), 4C (Q6, 13, 14, 15, 16), 4D (Q2, 8, 11, 12, 14).
- Practice expressing sums of sines and cosines in r-form.
- Memorize exact values for sine, cosine, and tangent at standard angles.
- Work through assigned trigonometric proofs and maximum/minimum value problems.