📏 trigonometry

1. **State the problem:** A boy observes a tree from two points on the ground. From the first point, the angle of elevation to the top of the tree is $25^\circ$. He then walks 30 m

1. **State the problem:** Prove that $$\frac{\sin^2 x - 1}{\tan x \sin x - \tan x} = \cos x + \cot x.$$\n\n2. **Rewrite the numerator:** Note that $$\sin^2 x - 1 = -(1 - \sin^2 x)

1. **State the problem:** Prove that $$\frac{1+2\cos x \sin x}{\cos x \sin x + \cos^2 x} = 1 + \tan x.$$\n\n2. **Rewrite the expression:** Start with the left-hand side (LHS): $$\f

1. **State the problem:** Simplify the expression $$\frac{1+2\cos x \sin x}{\cos x \sin x + \cos^2 x}$$. 2. **Rewrite the numerator:** The numerator is $$1 + 2\cos x \sin x$$.

1. **State the problem:** Use De Moivre's theorem to show that $$\frac{\sin 4\theta}{\sin \theta} = 8 \cos 3\theta - 4 \cos \theta.$$\n\n2. **Recall De Moivre's theorem:** For any

1. **State the problem:** Simplify the expression $$\frac{1-2\sin x \cos x}{1+2\sin x \cos x}$$. 2. **Recall the double-angle identity:** We know that $$\sin(2x) = 2\sin x \cos x$$

Problem: In triangle ABC, the angles satisfy $A=70^\circ$, $B=50^\circ$, and the side length $c=18$. Find the side $a$ to two decimal places. 1. The sum of the angles in a triangle

1. The problem is to solve the equation $2 \sin(x - 1) = 0$ for $x$. 2. First, divide both sides of the equation by 2 to isolate the sine function:

1. Simplify the expression \(\sec^2 x |\cot^2 x - \cos^2 x|\). Recall that \(\sec^2 x = \frac{1}{\cos^2 x}\) and \(\cot^2 x = \frac{\cos^2 x}{\sin^2 x}\).

1. The problem involves understanding the position of a seat on a Ferris wheel at a 20° angle and interpreting coterminal and reference angles. 2. A Ferris wheel rotates in a circl

1. The problem involves understanding angles on a circle, specifically coterminal angles and reference angles, as applied to a Ferris wheel. 2. Coterminal angles are angles that di

1. **Problem Statement:** Analyze the rotation of a Ferris wheel seat using coterminal and reference angles for five given rotation angles.

1. The problem involves understanding angles on a Ferris wheel, their positions, and coterminal angles. 2. The Ferris wheel rotates counterclockwise, starting at 0° on the rightmos

1. **State the problem:** We have a viewing tower 30 meters above the ground. The angle of depression to an object on the ground is 30 degrees, and the angle of elevation to an air

1. **State the problem:** Simplify and verify the expression $$\frac{\sin^4 x + \cos^4 x - 1}{\sin^6 x + \cos^6 x - 1} = \frac{2}{3}$$. 2. **Simplify the numerator:**

1. The problem is to find the value of $y$ given the equation $(\sin x + \cos x) y = \cos^2 x$ at $x = \frac{\pi}{2}$. 2. Substitute $x = \frac{\pi}{2}$ into the equation:

1. **State the problem:** Given that $\sin \theta = \frac{\sqrt{3}}{2}$ and $\theta$ is acute, find $\tan 2\theta$ in surd form. 2. **Identify $\theta$:** Since $\sin \theta = \fra

1. **State the problem:** We want to prove the trigonometric identity: $$\cos 6x + \cos 4x \equiv 2 \cos 5x \cos x.$$

1. Résoudre dans ]-\pi ; \pi] : 1.a. Trouver $x$ tel que $\cos(x) = \frac{1}{2}$ et $\sin(x) = -\frac{\sqrt{3}}{2}$.

1. **State the problem:** Simplify the expression $$\frac{3\tan(330^\circ)\sec(120^\circ)}{4\csc(210^\circ)}$$ and convert degrees to radians first. 2. **Convert degrees to radians

1. **State the problem:** Solve for $x$ in the interval $0 \leq x \leq 360$ where $$\cos 3x - \cos x = 0.$$\n\n2. **Rewrite the equation:** We want to find $x$ such that $$\cos 3x