Grade 11 Trigonometric Equations quiz

📚 Quick Study Guide

📐 Understanding Trigonometric Ratios: Remember SOH CAH TOA! Sine ($\sin$) is Opposite/Hypotenuse, Cosine ($\cos$) is Adjacent/Hypotenuse, and Tangent ($\tan$) is Opposite/Adjacent. 🔄 Reciprocal Trigonometric Functions: Cosecant ($\csc$) is the reciprocal of Sine, Secant ($\sec$) is the reciprocal of Cosine, and Cotangent ($\cot$) is the reciprocal of Tangent. ➕ Trigonometric Identities: Key identities include $\sin^2(\theta) + \cos^2(\theta) = 1$, $1 + \tan^2(\theta) = \sec^2(\theta)$, and $1 + \cot^2(\theta) = \csc^2(\theta)$. 🧭 General Solutions: For equations like $\sin(x) = a$, find the principal value and then use the general solution formulas considering the periodicity of trigonometric functions. 📈 Solving Trigonometric Equations: Use algebraic manipulation to isolate the trigonometric function and then find the angles that satisfy the equation within the given interval. 💡 Double Angle Formulas: Know formulas such as $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$ and $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$.

🤔 Practice Quiz

1. What is the general solution for the equation $\sin(x) = 0$?
   1. $x = n\pi$, where n is an integer
   2. $x = 2n\pi$, where n is an integer
   3. $x = \frac{n\pi}{2}$, where n is an integer


2. Solve for $x$ in the equation $2\cos(x) - 1 = 0$ within the interval $[0, 2\pi]$.
   1. $x = \frac{\pi}{3}, \frac{5\pi}{3}$
   2. $x = \frac{\pi}{6}, \frac{11\pi}{6}$
   3. $x = \frac{\pi}{4}, \frac{7\pi}{4}$


3. Simplify the expression $\frac{\sin(2x)}{\sin(x)}$.
   1. $2\cos(x)$
   2. $2\sin(x)$
   3. $\cos(2x)$


4. What is the value of $\tan(\frac{\pi}{4})$?
   1. 1
   2. 0
   3. $\sqrt{3}$


5. Solve for $x$: $\cos^2(x) = 1$.
   1. $x = n\pi$, where n is an integer
   2. $x = \frac{n\pi}{2}$, where n is an integer
   3. $x = 2n\pi$, where n is an integer


6. If $\sin(\theta) = \frac{1}{2}$, what could be a possible value of $\theta$ in degrees?
   1. 30°
   2. 45°
   3. 60°


7. Simplify: $\sin^2(x) + \cos^2(x) + \tan^2(x)$.
   1. $\sec^2(x)$
   2. $\csc^2(x)$
   3. 1