Core Concept
Trigonometric Identities are equations involving trigonometric functions that are true for all values in their domain. Mastering these identities is key to learning trigonometry.
Fundamental Identities
Pythagorean Identities
sin 2 α + cos 2 α = 1 \sin^2\alpha + \cos^2\alpha = 1 sin 2 α + cos 2 α = 1 1 + tan 2 α = sec 2 α 1 + \tan^2\alpha = \sec^2\alpha 1 + tan 2 α = sec 2 α
Quotient Identities
tan α = sin α cos α \tan\alpha = \frac{\sin\alpha}{\cos\alpha} tan α = c o s α s i n α
Reduction Formulas
Negative Angles
sin ( − α ) = − sin α \sin(-\alpha) = -\sin\alpha sin ( − α ) = − sin α cos ( − α ) = cos α \cos(-\alpha) = \cos\alpha cos ( − α ) = cos α
Complementary Angles
sin ( π 2 − α ) = cos α \sin\left(\frac{\pi}{2} - \alpha\right) = \cos\alpha sin ( 2 π − α ) = cos α cos ( π 2 − α ) = sin α \cos\left(\frac{\pi}{2} - \alpha\right) = \sin\alpha cos ( 2 π − α ) = sin α
Supplementary Angles
sin ( π − α ) = sin α \sin(\pi - \alpha) = \sin\alpha sin ( π − α ) = sin α cos ( π − α ) = − cos α \cos(\pi - \alpha) = -\cos\alpha cos ( π − α ) = − cos α
Sum and Difference Formulas
Sine
sin ( α ± β ) = sin α cos β ± cos α sin β \sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta sin ( α ± β ) = sin α cos β ± cos α sin β
Cosine
cos ( α ± β ) = cos α cos β ∓ sin α sin β \cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta cos ( α ± β ) = cos α cos β ∓ sin α sin β
Tangent
tan ( α ± β ) = tan α ± tan β 1 ∓ tan α tan β \tan(\alpha \pm \beta) = \frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha\tan\beta} tan ( α ± β ) = 1 ∓ t a n α t a n β t a n α ± t a n β
Double Angle Formulas
Sine
sin 2 α = 2 sin α cos α \sin 2\alpha = 2\sin\alpha\cos\alpha sin 2 α = 2 sin α cos α
Cosine
cos 2 α = cos 2 α − sin 2 α = 2 cos 2 α − 1 = 1 − 2 sin 2 α \cos 2\alpha = \cos^2\alpha - \sin^2\alpha = 2\cos^2\alpha - 1 = 1 - 2\sin^2\alpha cos 2 α = cos 2 α − sin 2 α = 2 cos 2 α − 1 = 1 − 2 sin 2 α
Power-Reducing Formulas :
cos 2 α = 1 + cos 2 α 2 \cos^2\alpha = \frac{1 + \cos 2\alpha}{2} cos 2 α = 2 1 + c o s 2 α sin 2 α = 1 − cos 2 α 2 \sin^2\alpha = \frac{1 - \cos 2\alpha}{2} sin 2 α = 2 1 − c o s 2 α
Tangent
tan 2 α = 2 tan α 1 − tan 2 α \tan 2\alpha = \frac{2\tan\alpha}{1 - \tan^2\alpha} tan 2 α = 1 − t a n 2 α 2 t a n α
Auxiliary Angle Formula
a sin x + b cos x = a 2 + b 2 sin ( x + φ ) a\sin x + b\cos x = \sqrt{a^2 + b^2}\sin(x + \varphi) a sin x + b cos x = a 2 + b 2 sin ( x + φ )
where tan φ = b a \tan\varphi = \frac{b}{a} tan φ = a b
CSCA Practice Problems
[Example 1] Basic (Difficulty ★★☆☆☆)
Simplify: sin 2 α + cos 2 α + tan 2 α \sin^2\alpha + \cos^2\alpha + \tan^2\alpha sin 2 α + cos 2 α + tan 2 α
Solution :
sin 2 α + cos 2 α + tan 2 α = 1 + tan 2 α = sec 2 α \sin^2\alpha + \cos^2\alpha + \tan^2\alpha = 1 + \tan^2\alpha = \sec^2\alpha sin 2 α + cos 2 α + tan 2 α = 1 + tan 2 α = sec 2 α
Answer : sec 2 α \sec^2\alpha sec 2 α
[Example 2] Intermediate (Difficulty ★★★☆☆)
Given sin α = 3 5 \sin\alpha = \frac{3}{5} sin α = 5 3 α ∈ ( 0 , π 2 ) \alpha \in (0, \frac{\pi}{2}) α ∈ ( 0 , 2 π ) sin 2 α \sin 2\alpha sin 2 α
Solution :
cos α = 1 − sin 2 α = 4 5 \cos\alpha = \sqrt{1 - \sin^2\alpha} = \frac{4}{5} cos α = 1 − sin 2 α = 5 4
sin 2 α = 2 sin α cos α = 2 × 3 5 × 4 5 = 24 25 \sin 2\alpha = 2\sin\alpha\cos\alpha = 2 \times \frac{3}{5} \times \frac{4}{5} = \frac{24}{25} sin 2 α = 2 sin α cos α = 2 × 5 3 × 5 4 = 25 24
Answer : 24 25 \frac{24}{25} 25 24
Common Misconceptions
❌ Misconception 1: Wrong reduction formula
Wrong : sin ( π − α ) = − sin α \sin(\pi - \alpha) = -\sin\alpha sin ( π − α ) = − sin α
Correct : sin ( π − α ) = sin α \sin(\pi - \alpha) = \sin\alpha sin ( π − α ) = sin α
❌ Misconception 2: Confusing sum formulas
Wrong : sin ( α + β ) = sin α + sin β \sin(\alpha + \beta) = \sin\alpha + \sin\beta sin ( α + β ) = sin α + sin β
Correct : sin ( α + β ) = sin α cos β + cos α sin β \sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta sin ( α + β ) = sin α cos β + cos α sin β
Study Tips
✅ Categorize : Basic, reduction, sum/difference, double angle
✅ Understand derivation : Don't just memorize
✅ Practice : Reinforce through exercises
✅ Connect formulas : Many can be derived from each other
💡 Exam Tip : Trigonometric identities are core to trigonometry, mandatory in CSCA! Mastering formulas is key to problem-solving.