📏 trigonometry

1. **Problem:** Determine which statement about sine function is true for any real number $x$. 2. **Recall:** The sine function satisfies the inequality $$-1 \leq \sin t \leq 1$$ f

1. **Problem statement:** Find $\tan(x)$ for a right triangle with legs 3 (vertical) and 4 (horizontal), and hypotenuse 5, where angle $x$ is adjacent to the side of length 4. 2. *

1. **State the problem:** We need to solve for $x$ in the equation $$2x = 2n\pi \pm \frac{\pi}{6} - \frac{\pi}{3}$$ where $n$ is an integer. 2. **Rewrite the equation:** The right

1. We are asked to find $\sin(c)$ in a right triangle where the hypotenuse is 75, the side opposite angle $c$ is 21, and the adjacent side to angle $c$ is 72. 2. Recall the definit

1. Énoncé du problème : Déterminer les mesures des angles BAC et ACB dans un triangle rectangle ABC avec l'angle droit en B, où AC = 51 et BC = 48.

1. **Problem 1:** Given a right triangle with angle $\theta$, opposite side = 9, adjacent side = 40. 2. **Find:** $\sin \theta$, $\csc \theta$, $\cos \theta$, $\sec \theta$, $\tan

1. **Problem statement:** An observer in a boat sees a tower on top of a cliff. The angle of elevation to the tower's top is 60° and to the cliff's top is 30°. The tower's height i

1. **State the problem:** Solve the trigonometric equation $$2\sin^2 x - \sin x - 1 = 0$$ for $$x \in [0, \pi]$$ by factoring it as a quadratic. 2. **Rewrite the equation:** Let $$

1. The problem is to simplify the expression $\sin^3 x \cdot \csc^2 x + \tan x \cdot \cos x$. 2. Recall the identities:

1. نبدأ ببيان المشكلة: لدينا زاويتان $\theta$ و $\alpha$ بحيث أن $\theta + \alpha = 90^\circ$. المطلوب هو حساب قيمة $\sin^2 \theta + \sin^2 \alpha$.\n\n2. نستخدم العلاقة المعروفة ف

1. The problem is to find the exact value of $\sin\left(-\frac{\pi}{12}\right)$.\n\n2. We use the identity for sine of a negative angle: $$\sin(-x) = -\sin(x)$$\n\n3. So, $$\sin\le

1. The problem is to verify the exact values of trigonometric functions for angles $-\frac{\pi}{12}$, $\frac{\pi}{12}$, and $\frac{5\pi}{12}$.\n\n2. We use the angle sum and differ

1. **State the problem:** We are given the function $$f(x) = (\tan x)^{2020} + (\tan x)^{2022}$$ defined on the interval $$I = \left]-\frac{\pi}{2}, \frac{\pi}{2}\right[.$$ We want

1. The problem is to find the range of the function $y = 3\sin(\theta)$.\n\n2. Recall that the sine function $\sin(\theta)$ has a range of $[-1, 1]$. This means $\sin(\theta)$ can

1. The problem is to simplify or evaluate the expression $\cos \frac{\pi}{2n+1}$. 2. The cosine function $\cos x$ is defined for all real numbers $x$ and is periodic with period $2

1. **State the problem:** Evaluate the expression $\sin(420^\circ) \cos(390^\circ) + \cos(-300^\circ) \sin(-330^\circ)$.\n\n2. **Recall the formula:** This expression resembles the

1. **Problem:** Sketch a rotation angle $\theta$ in the second quadrant with a reference angle of $70^\circ$. Then state the measures of $\theta$ and another positive angle less th

1. **State the problem:** We need to solve for $x$, the length of side CE in right triangle CDE, where angle $C = 17^\circ$, angle $D = 90^\circ$, and side DE (opposite angle $C$)

1. Сформулюємо задачу: спростити вираз $$ (1 + \cot x)^2 + (1 - \cot x)^2 $$.\n\n2. Використаємо формулу квадрата суми та різниці: $$ (a+b)^2 = a^2 + 2ab + b^2 $$ і $$ (a-b)^2 = a^

1. Сформулюємо задачу: потрібно розв'язати рівняння $\sin^2 x = 1$. 2. Використаємо основне тригонометричне правило: $\sin^2 x = 1$ означає, що $\sin x = \pm 1$.

1. **Problem 1:** Determine an equation for the sinusoidal function given the data points: x: 0°, 45°, 90°, 135°, 180°, 225°, 270°