📏 trigonometry

1. مسئله: در نمودار تابع $y=\tan x$، نقاط $a$ و $b$ روی نمودار مشخص شده‌اند. مقدار $a-b$ را بیابید. 2. تابع تانژانت تعریف شده است به صورت:

1. مسئله: مقدار تابع $$y = a \sin \pi \left( \frac{1}{3}x - b \right) + c$$ را برای $$x = \frac{7}{3}$$ پیدا کنیم. 2. ابتدا باید مقادیر $$a$$، $$b$$ و $$c$$ را از نمودار تشخیص دهیم

1. مسئله: مقدار تابع $$f(x) = a \cos(\pi + bx)$$ را در نقطه $$x = -\frac{22}{3}$$ پیدا کنیم. 2. از نمودار می‌بینیم که دامنه (amplitude) موج برابر 4 است، پس $$a = 4$$.

1. **State the problem:** We need to find the least possible height of the post AB such that the zip wire AC makes an angle $x \geq 5^\circ$ with the horizontal. 2. **Given:**

1. **State the problem:** We have a right triangle formed by points A, B, C, and D with a zip wire AC as the hypotenuse. Given CD = 2.6 m, BD = 12 m, and BC calculated as $\sqrt{12

1. **Problem Statement:** We are asked to substitute $\cos(\theta_L)$ and $\cos(\theta_w)$ into the equation for $\frac{dT}{dx}$ to find a relation between these cosines. 2. **Unde

1. **State the problem:** We need to find the angle $\theta$ at vertex B in the first triangle using the Law of Cosines. The sides are given as follows: side opposite $\theta$ is 4

1. **State the problem:** We are given a triangle with one side of length 400 opposite an angle of 98.4° and another side opposite an angle of 24.6°. We need to find the length $x$

1. **Problem 6:** Use the Law of Cosines to find angle $\theta$ at vertex B in a triangle with sides $AB=42.75$, $BC=64.01$, and $AC=35.51$. 2. The Law of Cosines formula for angle

1. **Problem Statement:** Given the trigonometric values for angles A, B, C, D, E, and F, find the related angles in degrees and determine the quadrants where these angles lie. Als

1. **State the problem:** We need to find the exact value of $\cos(\alpha + \beta)$ given that $\tan \alpha = \frac{12}{5}$ with $\alpha$ in quadrant III, and $\cos \beta = \frac{1

1. **State the problem:** Given \( \cos \alpha = -\frac{12}{13} \) with \( \alpha \) in quadrant II, and \( \sin \beta = -\frac{8}{17} \) with \( \beta \) in quadrant III, find \(

1. **State the problem:** We need to find $\tan(\alpha + \beta)$ given $\tan \alpha = \frac{3}{4}$ with $\alpha$ in quadrant I, and $\cos \beta = \frac{4}{5}$ with $\beta$ in quadr

1. **State the problem:** We need to find $\sin(\alpha - \beta)$ given $\cos \alpha = \frac{12}{13}$ with $\alpha$ in quadrant IV, and $\tan \beta = \frac{5}{12}$ with $\beta$ in q

1. **Problem Statement:** Given the functions $f(x) = \cos 2x$ and $g(x) = \tan x + 2$ for $x \in [-180^\circ, 90^\circ]$, we need to sketch both graphs on the same axes, label int

1. **Problem Statement:** Given the functions $f(x) = \cos 2x$ and $g(x) = \tan x + 2$ for $x \in [-180^\circ, 90^\circ]$, we need to:

1. **Verify** $\csc \theta \sec \theta = \tan \theta + \cot \theta$. - Recall definitions: $\csc \theta = \frac{1}{\sin \theta}$, $\sec \theta = \frac{1}{\cos \theta}$, $\tan \thet

1. **Problem:** Verify the identity $$\csc \theta \sec \theta = \tan \theta + \cot \theta$$ - Recall definitions: $$\csc \theta = \frac{1}{\sin \theta}, \quad \sec \theta = \frac{1

1. **State the problem:** Solve the equation $$2 \cos^2 x = 1$$ on the interval $$[0, 2\pi)$$. 2. **Rewrite the equation:** Divide both sides by 2 to isolate $$\cos^2 x$$:

1. **Problem Statement:** You are at the top of a watchtower 100 feet above sea level. The angle of depression to a ship in the water is 25 degrees. You need to find the distance $

1. **Problem Statement:** Find the angle of elevation $\theta$ of the top of a flagpole that is $10\sqrt{3}$ meters tall from a point on the ground $30$ meters away from the base o