📏 trigonometry

1. **State the problem:** Solve the equation $$2 \cos^2 x - 3 \sin x = 3$$ for $$0 \leq x \leq 2\pi$$. 2. **Rewrite the equation using the Pythagorean identity:** Recall that $$\co

1. The problem states: Given $\cos a = \frac{5}{7}$ and the terminal side of angle $a$ lies in a certain quadrant, find the quadrant and evaluate $\cos\left(-\frac{a}{2}\right)$ us

1. **State the problem:** Autumn spots a plane flying at a constant altitude of 6950 feet. She measures the angle of elevation to the plane as 15° at point A and later as 38° at po

1. **State the problem:** Autumn spots a plane flying at a constant altitude of 69506950 feet. She measures the angle of elevation to the plane as 15° at point A and later as 38° a

1. Problem: Simplify each expression involving trigonometric functions of angle $\alpha$. 2. a) $1 - \cos^2 \alpha - \sin^2 \alpha = 1 - (\cos^2 \alpha + \sin^2 \alpha) = 1 - 1 = 0

1. We are given the function $g(\theta) = \sqrt{3} \cos \theta - \sin \theta$ and need to show that it can be rewritten as $g(\theta) = 2 \cos \left( \theta + \frac{\pi}{6} \right)

1. **State the problem:** Solve the equation $$\tan^2 x - \tan x = 0$$ for $$-\pi < x < \pi$$. 2. **Rewrite the equation:** Factor the left side:

1. **State the problem:** Show that $$\frac{1-\cos 2\alpha}{\sin 2\alpha} \equiv \tan \alpha.$$\n\n2. **Recall double-angle identities:**\n$$\cos 2\alpha = 1 - 2\sin^2 \alpha$$\n$$

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

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

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

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

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

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

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

1. The term \textbf{arctan} refers to the \textbf{inverse tangent function}.\n\n2. It is denoted as $\arctan(x)$ and it gives the angle whose tangent is $x$.\n\n3. More formally, i

1. Problem: Verify if the identity $\csc x \cdot \tan x / \sec x = 1$ is true. Step 1: Express all functions in terms of $\sin x$ and $\cos x$.

1. Problem: Find which expression is equivalent to $\sec^2 x$. Step 1: Recall the Pythagorean identity: $\sec^2 x = 1 + \tan^2 x$.