Trigonometric Identities
Learn the core trig identities β Pythagorean, reciprocal, and quotient β and use them to simplify expressions and verify equations without plugging in numbers.
Trig identities are algebraic relationships between sin, cos, tan and their relatives that are always true β for every angle. They are the rewrite rules of trigonometry. Engineers use them to simplify circuit equations. Calculus students use them to evaluate integrals. Physicists use them to derive wave equations. The ability to transform a trig expression from one form into another is one of the most powerful mathematical skills you can develop β and it starts with the three families of identities introduced here.
- State and apply the Pythagorean identity \(\sin^2\theta + \cos^2\theta = 1\) and its two variants
- Apply the reciprocal identities (csc, sec, cot) and the quotient identities (tan, cot)
- Simplify trig expressions by substituting equivalent forms
- Verify trig identities by transforming one side to match the other
Pythagorean Identities (3 forms β memorize all three):
\[ \sin^2\theta + \cos^2\theta = 1 \qquad 1 + \tan^2\theta = \sec^2\theta \qquad 1 + \cot^2\theta = \csc^2\theta \]Reciprocal Identities:
\[ \csc\theta = \frac{1}{\sin\theta} \qquad \sec\theta = \frac{1}{\cos\theta} \qquad \cot\theta = \frac{1}{\tan\theta} \]Quotient Identities:
\[ \tan\theta = \frac{\sin\theta}{\cos\theta} \qquad \cot\theta = \frac{\cos\theta}{\sin\theta} \]Derivation of the Pythagorean Identity
On the unit circle, any point is \((\cos\theta, \sin\theta)\) at distance 1 from the origin.
By the Pythagorean Theorem: \(x^2 + y^2 = r^2\)
Substituting: \(\cos^2\theta + \sin^2\theta = 1^2 = 1\) β
Dividing both sides by \(\cos^2\theta\): \(1 + \tan^2\theta = \sec^2\theta\) β
Dividing both sides by \(\sin^2\theta\): \(\cot^2\theta + 1 = \csc^2\theta\) β
Simplify: \(1 - \sin^2\theta\)
\[ 1 - \sin^2\theta = \cos^2\theta \quad \text{(from } \sin^2\theta + \cos^2\theta = 1 \text{)} \]Simplify: \(\dfrac{\sin\theta}{\cos\theta} \cdot \cos^2\theta\)
\[ \frac{\sin\theta}{\cos\theta} \cdot \cos^2\theta = \sin\theta \cdot \cos\theta = \frac{\sin(2\theta)}{2} \]Step 1: Cancel one cos ΞΈ. Step 2: The result sin ΞΈ cos ΞΈ can also be written as Β½ sin(2ΞΈ) using the double-angle identity.
Verify: \(\tan\theta \cdot \cos\theta = \sin\theta\)
Work on the left side only:
\[ \tan\theta \cdot \cos\theta = \frac{\sin\theta}{\cos\theta} \cdot \cos\theta = \sin\theta \quad \checkmark \]Rule: When verifying identities, transform one side only (usually the more complex side) until it equals the other. Never move terms across the equal sign.
- Simplify \(\sin^2\theta + \cos^2\theta + \tan^2\theta\) using identities.
- If \(\sin\theta = 3/5\), find \(\cos\theta\) (assume Quadrant I).
- Verify: \(\cot\theta \cdot \sin\theta = \cos\theta\).
βΆ Show Answers
- \(1 + \tan^2\theta = \sec^2\theta\).
- \(\cos\theta = \sqrt{1 - (9/25)} = \sqrt{16/25} = \mathbf{4/5}\).
- \(\cot\theta \cdot \sin\theta = \frac{\cos\theta}{\sin\theta} \cdot \sin\theta = \cos\theta\) β
- Treating \(\sin^2\theta\) as \(\sin(\theta^2)\): The notation \(\sin^2\theta\) means \((\sin\theta)^2\) β square the function value, not the angle.
- Moving terms across the equal sign when verifying: Verification means showing LHS = RHS by transforming one side only. Cross-multiplying or adding to both sides is not valid in a proof.
- Forgetting to check where an identity is undefined: \(\tan\theta = \sin\theta/\cos\theta\) is undefined when \(\cos\theta = 0\) (at 90Β°, 270Β°, etc.). Identities hold where the functions are defined.
- Pythagorean: \(\sin^2\theta + \cos^2\theta = 1\) and its two derived variants.
- Reciprocal: csc = 1/sin, sec = 1/cos, cot = 1/tan.
- Quotient: tan = sin/cos, cot = cos/sin.
- To verify: transform the complex side using substitution. Never cross the equals sign.
Trig identities are essential for integration in calculus. To evaluate an integral like \(\int \sin^2\theta \, d\theta\), you first rewrite \(\sin^2\theta = \frac{1 - \cos(2\theta)}{2}\) using the double-angle identity β and then integrate the simpler form. In physics, interference patterns, standing waves, and quantum mechanics wave functions all require trig identity manipulation to simplify the equations of motion into solvable forms.
Calculator Connection
The Trig Identity Verifier checks whether two expressions are equivalent for a range of angles. The Sum & Difference Formulas Calculator and Double/Half-Angle Formulas Calculator expand the core identity toolkit.
Try it with the Calculator
Apply what you've learned with these tools.
Trigonometric Identities (Basics) β Quiz
5 questions per attempt Β· Intermediate Β· Pass at 70%
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