📏 trigonometry

1. **Problem statement:** Find the solutions to the equation $\sin(x) = 0.4$ in the interval $[0^\circ; 360^\circ]$. 2. **Formula and rules:** The sine function is periodic with pe

1. **Problem Statement:** Evaluate the expression $$\arcsin\left(\frac{\sqrt{3}}{2}\right) + \arctan(1) - \arcsec(\sqrt{2})$$ and verify if it equals $$\frac{\pi}{4}$$. 2. **Recall

1. **Problem statement:** A 30 m ladder reaches a window 26 m high when placed at point A. After fixing the first window, the ladder is pushed back to point B, reducing the angle w

1. The problem is to express trigonometric functions \(\sin x\), \(\cos x\), and \(\tan x\) in terms of each other without using inverse functions. 2. Recall the fundamental identi

1. **Problem statement:** Solve the following trigonometric equations for $x$. 2. **Formula and rules:**

1. We are asked to solve the following trigonometric equations for $x$: (i) $\sin x = \frac{1}{2}$

1. **Problem statement:** Prove that $$\tan \theta + \cot \theta \equiv 2 \csc 2\theta$$ for $$\theta \neq \frac{n\pi}{2}, n \in \mathbb{Z}$$. 2. **Recall definitions and identitie

1. **Problem statement:** Solve the following trigonometric equations for $x$. 2. **Recall the general solutions:**

1. We are asked to solve the trigonometric equations for $x$ where $\sin x$, $\cos x$, or $\tan x$ equals given values. 2. The general solutions for sine, cosine, and tangent equat

1. **Problem Statement:** Find the period, amplitude, constants affecting the function, domain, and range of the sine function given in part 2a and graph the function in part 2b.

1. The problem is to analyze the function $y = 2\sin(5x^2 + 4)$.\n\n2. The general form of a sine function is $y = A\sin(Bx + C)$, where $A$ is the amplitude, $B$ affects the perio

1. **State the problem:** We need to analyze the function $y = 2\sin(5x^2 + 4)$.\n\n2. **Formula and rules:** The sine function $\sin(\theta)$ oscillates between $-1$ and $1$. Mult

1. **State the problem:** Find the solutions of the equation $$\sin 4\theta + 2 \sin \theta \cos \theta = 0.$$\n\n2. **Recall formulas:** Use the double-angle and multiple-angle id

1. The problem is to simplify the expression $3 - 4\cos(\theta)$, where $\theta$ is an angle. 2. The expression involves a constant term and a cosine function multiplied by a const

1. **State the problem:** From a point on the ground 15 meters from the base of a tree, the angle of elevation to the top of the tree is 35°. We need to find the height of the tree

1. المشكلة: حساب قيمة $\tan 2x$. 2. القاعدة المستخدمة: صيغة الزاوية المزدوجة للظل هي

1. The problem is to solve for $x$ given the equation $\cos x = -\frac{2}{7}$.\n\n2. Recall that the cosine function, $\cos x$, gives the ratio of the adjacent side to the hypotenu

1. **Stating the problem:** Simplify the expression $$\frac{\sin^2 x - \cos^2 x}{\cos x \sin x}$$ and relate it to the identity $$\frac{\sin^2 x + \cos^2 x}{\cos x \sin x}$$. 2. **

1. **Problem statement:** Prove the identity $$\frac{1+\tan x}{1+\cot x} \equiv \frac{1+\tan x}{\cot x -1}$$

1. **State the problem:** Given polar coordinates $\theta = 115^\circ$ and $r = 25$, convert these to Cartesian coordinates $(x, y)$. 2. **Formula used:** The conversion from polar

1. **Restating the problem:** Find the exact value of $\tan\left(\frac{\pi}{3}\right)$.\n\n2. **Formula used:** $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$. This means tange