📏 trigonometry

1. **State the problem:** Solve the equation $\sin x = \sin 282^\circ$ for $0^\circ \leq x \leq 270^\circ$. 2. **Recall the sine identity:** For angles $x$ and $a$, $\sin x = \sin

1. The problem asks us to find the values of $x$ in the interval $180^\circ \leq x \leq 360^\circ$ such that $\cos x = \cos 132^\circ$. 2. Recall the cosine function's property: $\

1. **State the problem:** Solve the equation $\tan x = \tan 38^\circ$ for $90^\circ \leq x \leq 360^\circ$. 2. **Recall the periodicity of tangent:** The tangent function has a per

1. Express $\tan 4x$ in terms of $\tan x$. Using the tangent multiple angle formula:

1. **State the problem:** Show that $ (\sin A + \cos A)^2 = 1 + \sin 2A $ and find the maximum value of $ 4(\sin A + \cos A)^2 $. 2. **Expand the left side:**

1. **State the problem:** Solve the equation $$15 \sin^2 x = 13 + \cos x$$ for $$0^\circ \leq x \leq 180^\circ$$. 2. **Rewrite the equation:** Use the Pythagorean identity $$\sin^2

1. **State the problem:** (i) Given the equation $$3 \sin^2 x - 8 \cos x - 7 = 0,$$ show that for real values of $$x,$$ $$\cos x = -\frac{2}{3}.$$

1. Problem 26(i): Prove the identity $$\frac{\cos \theta}{\tan \theta (1 - \sin \theta)} \equiv 1 + \frac{1}{\sin \theta}$$ for all valid $\theta$. 2. Start with the left-hand side

1. Problem Q.62: Simplify $$\sqrt{\sec^2\theta + \csc^2\theta} \times \sqrt{\tan^2\theta - \sin^2\theta}$$. Step 1: Use identities: $$\sec^2\theta = 1 + \tan^2\theta$$ and $$\csc^2

1. **State the problem:** Simplify the expression $$\frac{\cos(\theta)}{1 - \sin(\theta)} - \tan(\theta)$$. 2. **Rewrite the tangent term:** Recall that $$\tan(\theta) = \frac{\sin

1. **State the problem:** Solve for $\theta$ in degrees the equation $$2\sin^2(\theta) - 3\sin(\theta) + 1 = 0$$ where $0^\circ \leq \theta < 360^\circ$. 2. **Substitute:** Let $x

1. The problem is to evaluate $\sin\left(\frac{\pi}{2}\right)$.\n\n2. Recall that $\sin(\theta)$ gives the y-coordinate of the point on the unit circle at angle $\theta$.\n\n3. The

1. The term "Sin invers" usually refers to the inverse sine function, also called arcsine. 2. The inverse sine function is denoted as $\sin^{-1}(x)$ or $\arcsin(x)$.

1. **Problem Statement:** Find the expansions of $\sin(A+B+C)$, $\cos(A+B+C)$, and $\tan(A+B+C)$.\n\n2. **Expansion of $\sin(A+B+C)$:** Use the angle addition formula for sine twic

1. **Problem statement:** Find the expansions of $\sin(A + B + C)$, $\cos(A + B + C)$, and $\tan(A + B + C)$. 2. **Expand $\sin(A + B + C)$:** Use the angle sum formula for sine:

1. **State the problem:** Solve the equation $\tan x = \cot x + 2 \tan(2x - 90^\circ)$. 2. **Rewrite trigonometric functions:** Recall that $\cot x = \frac{1}{\tan x}$ and $\tan(2x

1. The problem is to calculate $\cos 75^\circ$ and round the result to 2 decimal places. 2. We can use the angle sum identity for cosine: $$\cos(a+b) = \cos a \cos b - \sin a \sin

1. **State the problem:** (i) Prove the identity $$\frac{\cos\theta}{\tan\theta(1-\sin\theta)} \equiv 1 + \sin\theta.$$

1. **State the problem:** Prove the identity \( \frac{\cos \theta}{\tan \theta} (1 - \sin \theta) \equiv 1 + \frac{1}{\sin \theta} \). Then solve \( \frac{\cos \theta}{\tan \theta}

1. The problem is to analyze the function $y = \sin 5x \cdot \cos 3x$. 2. We can use the product-to-sum identity for sine and cosine:

1. **State the problem:** We need to show that the equation $$2 \tan^2 \theta \sin^2 \theta = 1$$ can be rewritten as $$2 \sin^4 \theta + \sin^2 \theta - 1 = 0$$. 2. **Recall the i