📏 trigonometry

1. **Problem Statement:** Given that $\sec \theta = a + \frac{1}{4a}$, find the value of $\sec \theta + \tan \theta$. 2. **Recall the identity:**

1. **State the problem:** Solve the trigonometric equation $$56.640 \cos \theta - \sin \theta = -0.25$$ for $$\theta$$. 2. **Formula and approach:** We can rewrite the expression $

1. **State the problem:** Solve the trigonometric equation $$2\sin^2 x - 3\sin x + 1 = 0$$ for $x$. 2. **Identify the substitution:** Let $y = \sin x$. The equation becomes a quadr

1. **State the problem:** Solve the trigonometric equation $$2\sin^2 x - 3\sin x = -1$$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero

1. **Stating the problem:** We have a right triangle with a 90° angle and a 30° angle. The side opposite the 30° angle is 3, and the hypotenuse is $x$. We want to find $x$ using th

1. **Problem Statement:** We need to graph the function $y = \cos(3x + \frac{\pi}{2})$. 2. **Starting Graph:** The base graph is $y = \cos(x)$, which has a period of $2\pi$ and amp

1. The period of a function, especially for trigonometric functions like sine and cosine, is the length of one complete cycle of the wave. 2. For a function of the form $y = a \sin

1. Let's clarify the problem: You asked, "so what is the period?" This usually refers to the period of a periodic function, such as sine, cosine, or any repeating wave. 2. The peri

1. The problem is to graph the function $y = \cos(3x + \frac{\pi}{2})$. 2. We start with the basic cosine function $y = \cos x$, which has a period of $2\pi$ and amplitude 1.

1. Let's clarify the context of step 5 where $\alpha$ is divided by 2. 2. Often in trigonometry or geometry problems, dividing an angle $\alpha$ by 2 is related to using half-angle

1. **State the problem:** We need to understand the angle $\frac{5\pi}{4}$ radians on the unit circle and its position. 2. **Recall the unit circle basics:** The unit circle has ra

1. مسئله اول: تابع $f(x) = 5 \sin\left(3\left(\frac{\pi}{6}x - c\right)\right)$ در $x=\frac{1}{\gamma}$ ماکسیمم می‌شود. می‌خواهیم طول نقطه مینیمم آن را پیدا کنیم. 2. برای ماکسیمم ت

1. **Problem Statement:** We have the function $y = a + b \sin\left(x + \frac{\pi}{3}\right)$ and are given that the maximum value of $y$ is $\frac{\sqrt{3}}{2}$ and the minimum va

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 $x$