📏 trigonometry

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: $\sec^2 x = 1 + \tan^2 x$.

1. **State the problem:** Given the equation $\sec x - \tan x = 1$, find the value(s) of $x$. 2. **Recall trigonometric identities:** We know that $\sec x = \frac{1}{\cos x}$ and $

1. **State the problem:** Simplify the expression $2\cos^2 x \times \frac{\sin x}{\cos x}$.\n\n2. **Rewrite the expression:** The expression is $2\cos^2 x \cdot \frac{\sin x}{\cos

1. Problem 21: Simplify $$\cos(87\pi + \alpha) \cdot \cos\left(\frac{57\pi}{2} - \alpha\right) \cdot \tan(2025\pi + \alpha)$$. - Use periodicity: $$\cos(87\pi + \alpha) = \cos(\alp

1. The problem is to understand why $\cos(\pi + a)$ equals $-\cos(a)$.\n\n2. Recall the cosine addition formula: $$\cos(x + y) = \cos x \cos y - \sin x \sin y.$$\n\n3. Apply this f

1. The problem asks whether $\cos(87\pi + a)$ is positive or negative. 2. Recall the cosine addition formula and periodicity: cosine has period $2\pi$, so

1. We start with the equation: $$2\cos(2x) + 2\sin(2x) \cdot \cos(x) - 5\sin(x) \cdot \cos(2x) = 0$$ 2. Use double-angle identities: $$\cos(2x) = 2\cos^2(x) - 1$$ and $$\sin(2x) =

1. The problem is to understand the function $\sin x$ and its properties. 2. The sine function $\sin x$ is a periodic function with period $2\pi$, meaning $\sin(x + 2\pi) = \sin x$

1. **State the problem:** We are given the height function of a person on a Ferris wheel as $$h(t) = 20 \sin\left(\frac{\pi}{30}t - \frac{\pi}{2}\right) + 22$$ where $t$ is time in

1. **State the problem:** Solve the equation $$\tan \frac{1}{2}x = 3$$ for $$0 < x < 4\pi$$, giving answers in radians to 3 significant figures. 2. **Rewrite the equation:** Let $$

1. The problem states that angle $\theta$ is in quadrant IV and $\cos \theta = \frac{4}{5}$. We need to find $\tan \theta$. 2. Recall that in quadrant IV, cosine is positive and si

1. Problem: If angle $\theta$ is in quadrant IV and $\cos \theta = \frac{4}{5}$, find $\tan \theta$. 2. Since $\cos \theta = \frac{4}{5}$ and $\theta$ is in quadrant IV, $\sin \the

1. We are asked to verify the trigonometric identity: $$\sin 3A = \sin A (3 \cos^2 A - \sin^2 A)$$. 2. Start with the left-hand side (LHS): $$\sin 3A$$.

1. **State the problem:** Prove that $$\frac{1-\sin A}{1+\sin A} = (\sec A - \tan A)^2$$. 2. **Start with the right-hand side (RHS):**

1. Given that $\csc x = \frac{17}{15}$ and $x$ is a positive acute angle, we need to find the value of $x$. 2. Recall that $\csc x = \frac{1}{\sin x}$, so we have:

1. **State the problem:** Given $\tan x = -\frac{3}{4}$, find $\sin x$ and $\csc x$ for $x$ in the interval $0^\circ$ to $360^\circ$. 2. **Recall the identity:** $\tan x = \frac{\s

1. Given \( \tan x = \frac{3}{4} \), we need to find \( \sin x \) and \( \csc x \) for \( x \) in the interval \( 0^\circ \) to \( 360^\circ \). 2. Recall that \( \tan x = \frac{\s

1. **State the problem:** Prove that $$\frac{\tan x + \sec x - 1}{\tan x - \sec x + 1} = \frac{1 + \sin x}{\cos x}.$$\n\n2. **Rewrite tangent and secant in terms of sine and cosine