📏 trigonometry

1. **State the problem:** Prove that $$\frac{\sin x}{1-\cos x} + \frac{\tan x}{1+\cos x} = \cot x + \sec x \csc x$$. 2. **Recall formulas and identities:**

1. **Énoncé du problème :** Montrer que $0 < \arccos\left(\frac{3}{4}\right) < \frac{\pi}{4}$.

1. \textbf{المشكلة:} في دائرة الوحدة، نريد إثبات أن \sin \theta = y و \cos \theta = x حيث \theta هو الزاوية المرسومة في الدائرة. 2. \textbf{تعريف دائرة الوحدة:} دائرة الوحدة هي دائ

1. **Problem Statement:** Prove that on the unit circle, $\sin \theta = y$ and $\cos \theta = x$ where $(x,y)$ is a point on the circle corresponding to angle $\theta$.

1. **Problem statement:** Lara stands 40 meters from a tree and observes the tree's top at a 30° angle of elevation and the bottom of its reflection in the water at a 45° angle of

1. **Problem:** Calculate $\cos\left(\frac{\pi}{3}\right)$. 2. **Formula and rules:** The cosine of an angle in radians can be found using the unit circle or known special angles.

1. (a) Prove the identity $\cos 3\theta \equiv 4 \cos^3 \theta - 3 \cos \theta$ by expressing $3\theta$ as $2\theta + \theta$. Use the cosine addition formula: $\cos(a+b) = \cos a

1. **Problem (a):** Prove the identity $$\cos 3\theta \equiv 4 \cos^3 \theta - 3 \cos \theta$$ by expressing $$3\theta$$ as $$2\theta + \theta$$. 2. Use the cosine addition formula

1. Solve the equation $3 \cot x - 4 \cot 2x = 3$ for $0^\circ \leq x \leq 180^\circ$. 2. (a) Express $7 \sin \theta + 24 \cos \theta$ in the form $R \cos (\theta - \alpha)$, where

1. **Problem Statement:** A kite is flying at a height of 75 m from the ground, attached to a string inclined at 60° to the horizontal. Find the length of the string to the nearest

1. **State the problem:** Convert the Cartesian coordinate $(2, -6)$ to polar coordinates $(r, \theta)$ where $0 \leq \theta < 2\pi$. 2. **Recall the formulas:**

1. The problem asks to find the x-values of all vertical asymptotes of the function $$y = \csc(5x)$$ in the interval $$[0, 2\pi)$$. 2. Recall that $$\csc(\theta) = \frac{1}{\sin(\t

1. The problem is to evaluate the expression $2 \tan(\sin^{-1}(-1))$. 2. Recall that $\sin^{-1}(x)$ is the inverse sine function, which returns an angle $\theta$ such that $\sin(\t

1. The problem states: Given that $x + y = 90^\circ$, find the value of the expression $$\frac{\cos(2x + y) + \sin(3x + 2y)}{\cos y}$$

1. **Problem:** Given $\sec\theta + \tan\theta = p$, find the value of $\sin\theta$ in terms of $p$. 2. **Formula and rules:** Recall the identity:

1. **State the problem:** Simplify and verify the identity $$(\cot\theta + \csc\theta)(\tan\theta - \sin\theta) = \sec\theta - \cos\theta.$$\n\n2. **Recall definitions and formulas

1. **Problem:** Define the ASTC Rule. 2. **Explanation:** The ASTC Rule is a mnemonic to remember the signs of trigonometric functions in the four quadrants of the unit circle.

1. The problem is to understand or use the angle given, which is $36.9^\circ$. 2. Angles are measured in degrees and are used in various math and physics problems, such as trigonom

1. Problem: Evaluate $\cos(625)$.\n2. Formula and rules: Cosine is periodic with period $2\pi$, so $$\cos(x+2\pi k)=\cos x\text{ for any integer }k$$.\n3. Reduce the angle modulo $

1. **State the problem:** We need to find the value of $k$ in the equation $$\frac{\cot A}{1 + \csc A} - \frac{\cot A}{1 - \csc A} = \frac{k}{\cos A}.$$\n\n2. **Recall definitions

1. **Problem statement:** Find all values of $\theta$ such that $\tan \theta = 2$. 2. **Formula and rules:** The tangent function is periodic with period $\pi$, meaning if $\tan \t