📏 trigonometry

1. **State the problem:** We want to show that $\sin x + \sin y$ is equivalent to $2 \sin \left( \frac{x+y}{2} \right) \cos \left( \frac{x-y}{2} \right)$. 2. **Formula used:** This

1. **State the problem:** Evaluate $\sin(2\pi + 30)$. 2. **Recall the sine addition formula and periodicity:** The sine function has a period of $2\pi$, meaning $\sin(\theta + 2\pi

1. **Problem statement:** Prove the trigonometric identity $$\sin 4x \equiv 4 \sin x (2 \cos^3 x - \cos x)$$. 2. **Recall the double-angle formulas:**

1. **State the problem:** Given $\sin \theta = -\frac{\sqrt{7}}{5}$ and $\cos \theta > 0$, find $\cos \theta$ and $\tan \theta$. 2. **Recall the Pythagorean identity:**

1. Nyatakan nilai d dan c. Masalah ini tidak memberikan konteks langsung untuk nilai d dan c, jadi kita anggap ia berkaitan dengan persamaan atau fungsi yang diberikan.

1. **Problem statement:** Given $p = \frac{2}{a^2 + 1} - \frac{2}{b^2 + 1} + \frac{3}{c^2 + 1}$ with $a,b,c > 0$ and the condition $abc + a + c = b$, find the maximum value of $p$.

1. **Planteamiento del problema:** Resolver la ecuación $$\frac{2 \sin^2 2\theta - 5 \sin 2\theta - 3}{\sin 2\theta - 1} = 0$$. 2. **Interpretación:** La fracción es igual a cero c

1. The problem is to understand the basics of trigonometry for class X students. 2. Trigonometry deals with the relationships between the angles and sides of triangles, especially

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