Trigonometric Identities and Equations

trigonometricfunctions year11 advanced

Trigonometric identities and equations | NSW Curriculum Website

• MAV-11-05 uses periodic functions to solve trigonometric equations and prove trigonometric identities

📖 Prior Knowledge

Content	Prerequisite relationships
Trigonometry A	- Define the trigonometric ratios → Use complementary-angle relationships - Define the trigonometric ratios → Define sec, cosec and cot
Equations C	- Solve quadratic equations by factorisation → Solve trig equations reducible to quadratics
Trigonometry D	- Define sine and cosine using the unit circle → Define sec, cosec and cot - Define sine and cosine using the unit circle → Prove the Pythagorean identities - Express tangent as sine over cosine → Use the quotient identities - Use complementary-angle relationships (revisited content) - Use the related angle → Evaluate trig expressions from one ratio - Use the related angle → Solve trig equations on a restricted domain
Trigonometry and Measure of Angles	- Find exact ratios for multiples of $\frac{\pi }{6}$ and $\frac{\pi }{4}$ → Find exact sec, cosec and cot ratios - Find exact ratios for multiples of $\frac{\pi }{6}$ and $\frac{\pi }{4}$ → Solve trig equations on a restricted domain - Graph trig functions in radians → Solve trig equations on a restricted domain

Trigonometric identities and equations

• Define the trigonometric functions $\sec \theta$, $cosec\theta$ and $\cot \theta$ for acute angles $\theta$ using ratios of sides in a right-angled triangle
• Find the exact secant, cosecant and cotangent ratios for angles of 30°, 45° and 60°
• Justify the values of $\sec \theta$, $cosec\theta$ and $\cot \theta$ at 0° and 90° by examining the ratios of sides in a right-angled triangle as \theta tends to 0° and 90°
• Develop the definitions $\sec \theta =\frac{r}{x}$, $\mathrm{cosec}\theta =\frac{r}{y}$ and $\cot \theta =\frac{x}{y}$ using a circle of radius r centred at the origin
• Establish the reciprocal ratios $\sec A=\frac{1}{cosA}$, $\mathrm{cosec}A=\frac{1}{sinA}$, and $\cot A=\frac{1}{tanA}$, identifying the angles to be excluded in each identity
• Determine the exact value of the secant, cosecant and cotangent ratios for angles that are integer multiples of $\frac{\pi }{6}$ and $\frac{\pi }{4}$, if they exist
• Establish the identities $\tan \theta =\frac{\sin \theta }{\cos \theta }$ and $\cot \theta =\frac{\cos \theta }{\sin \theta }$, identifying the angles to be excluded in each identity
• Prove the complementary angle identities $\sin (90°-\theta )=\cos \theta ,$ $\cos (90°-\theta )=\sin \theta$, $\tan (90°-\theta )=\cot \theta$, $\cot (90°-\theta )=\tan \theta$, $\sec (90°-\theta )=\mathrm{cosec}\theta$ and $\mathrm{cosec}(90°-\theta )=\sec \theta$ identifying the angles to be excluded in each identity
• Evaluate trigonometric expressions using angles of any magnitude and complementary angle identities
• Solve equations involving trigonometric ratios of angles, specified in degrees or radians, on a restricted domain
• Prove the Pythagorean identity ${\cos }^{2}x+{\sin }^{2}x=1$, and the identities ${1+\tan }^{2}x={\sec }^{2}x$ and $1+{\cot }^{2}x={\mathrm{cosec}}^{2}x$
• Apply trigonometric identities to solve problems, simplify expressions and prove further trigonometric identities using substitution and/or reduction to $\sin x$ and $\cos x$
• Solve problems involving trigonometric equations, including those that reduce to quadratic equations, on a restricted domain