📏 trigonometry

1. **State the problem:** We have points A and B such that B is 10 cm due south of A.

1. **State the problem:** We need to find the angle $x^\circ$ in a right triangle where the adjacent side to $x$ is $38$ and the hypotenuse is $49$. 2. **Formula used:** In a right

1. **State the problem:** We have a right triangle with a vertical side of length $30$ cm, a base of length $15$ cm, and we want to find the angle $x$ between the slanted side (hyp

1. **State the problem:** We need to find the equation of a sinusoidal curve that starts at the point $(0, -1)$, rises to a peak near $(1.57, 0)$, and falls back to $(3.14, -1)$. 2

1. **State the problem:** We are given points and behavior of a wave-like function and need to find its equation. 2. **Analyze the graph:** The function starts at $y=3$ when $x=0$,

1. **State the problem:** Find the expression and properties of the function $\cos(x)$.\n\n2. **Formula and explanation:** The cosine function is defined as the ratio of the adjace

1. **State the problem:** Find the exact value of $$\cos(\alpha - \beta)$$ given $$\sin \alpha = \frac{7}{25}$$ with $$\frac{\pi}{2} < \alpha < \pi$$ and $$\cos \beta = \frac{2}{5}

1. **State the problem:** Solve the trigonometric equation $$\cos 5x + \sin x + \sin 7x - \cos 3x = 0$$. 2. **Use trigonometric identities:** Recall the sum-to-product formulas:

1. **Problem Statement:** We want to find the formula for a sinusoidal function that involves 3 amplitudes, 3 frequencies, and a time variable.

1. **State the problem:** We need to analyze the function $$y = -3 + 2\cot\left(\frac{\pi}{2} x + \frac{3\pi}{4}\right)$$ and identify the shape of its basic graph. 2. **Recall the

1. **State the problem:** We need to analyze and graph the function $$y = -2\csc\left(\frac{4}{7}x + 5\pi\right)$$ and determine how it is reflected, shifted, and stretched compare

1. The problem asks to resolve or solve a question involving the angle $\theta$ where $0 < \theta < 180$ degrees. 2. Since the exact equation or expression involving $\theta$ is no

1. **State the problem:** Solve the equation $$3\tan^2\theta = \frac{2}{\tan^2\theta}$$ for $$0 < \theta < x$$. 2. **Rewrite the equation:** Multiply both sides by $$\tan^2\theta$$

1. **State the problem:** We want to prove the identity $$\sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta$$ using double angle formulas. 2. **Recall the double angle formula for sin

1. **State the problem:** We want to sketch the graphs of $y=\sin x$ and $y=\sin 2x$ and understand the basic rules for these sine functions. 2. **Recall the basic sine function:**

1. **Problem Statement:** We need to solve a trigonometric expression or equation using only the sum and difference formulas, without using Simpson's rule or double and half-angle

1. **State the problem:** We want to verify or simplify the expression:

1. **State the problem:** We are given $\theta = 20^\circ$ and need to find $\alpha$ such that $0^\circ < \alpha < 360^\circ$, $\alpha \neq \theta$, and:

1. **State the problem:** We need to find the exact value of $\sin\left(\frac{\pi}{4}\right)$. 2. **Recall the formula and important rules:** The angle $\frac{\pi}{4}$ radians is e

1. **State the problem:** Prove that $$\cos^2(\theta + 45^\circ) - \frac{1}{2}(\cos 2\theta - \sin 2\theta) \equiv \sin^2 \theta.$$

1. **Énoncé du problème** : Montrer que pour tout entier naturel $p$ et tout réel $\theta \neq 0$, on a $$\frac{\sin((2p+1)\theta)}{\sin \theta} = 1 + 2 \sum_{k=1}^p \cos(2k\theta)