📏 trigonometry

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°

1. **Stating the problem:** Create and solve three trigonometric function problems. 2. **Problem 1:** Find $\sin(30^\circ)$.

1. The problem is to understand how to graph tangent functions. 2. The tangent function is defined as $y=\tan(x)$, which is the ratio of sine to cosine: $\tan(x) = \frac{\sin(x)}{\

1. The problem is to find the value of $\sin 50^\circ$. 2. The sine function gives the ratio of the length of the side opposite the angle to the hypotenuse in a right triangle.

1. The problem is to find the value of $\sin 30^\circ$. 2. The sine function relates an angle in a right triangle to the ratio of the length of the opposite side to the hypotenuse.

1. The problem is to find the value of $\sin 10^\circ$. 2. The sine function gives the ratio of the length of the side opposite the angle to the hypotenuse in a right triangle.

1. **State the problem:** We have two right-angled triangles ABD and BCD sharing altitude BD perpendicular to AC. Given AD = 5 m, DC = 14 m, and angle BAD = 53°, we need to find th

1. The problem asks to find an expression for $f(x)$ in the form $f(x) = a \cdot \sin(x - b)$ where $a$ is the amplitude and $b \in \mathbb{Z}$. 2. From the graph description, the

1. **State the problem:** An aircraft flies 500 km on a bearing of 100 degrees, then 600 km on a bearing of 160 degrees. We need to find the distance and bearing from the starting

1. **Problem statement:** A ship travels on a N 500 E course (meaning 50 degrees east of north). It travels until it is due north of a port that is 10 km due east of the starting p

1. **Problem Statement:** Solve the equation $\cos\theta = \frac{\theta}{2}$ for $\theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ graphically and find the approximate soluti

1. **State the problem:** A ship sails 200 km on a bearing of 243.7 degrees. We need to find how far south and how far west the ship has traveled. 2. **Understanding bearings:** Be

1. The problem involves finding the height of a tree using trigonometry. 2. Given: The distance from the instrument to the tree base is $9\sqrt{3}$ meters, and the angle of elevati