📏 trigonometry

1. **Problem:** Solve the equation $$12 \sin^2 x - 11 \sin x + 2 = 0$$ for $$\sin x$$. 2. **Formula and approach:** This is a quadratic equation in terms of $$\sin x$$. Let $$y = \

1. **Problem (i): Solve for $0 < \theta \leq 360^\circ$ the equation $4 \tan \theta + 5 \sin \theta = 0$.** 2. **Recall the definitions:**

1. The problem is to draw the function $y=\tan(x)$ in the interval $(-\pi, \pi)$.\n\n2. The tangent function is defined as $\tan(x) = \frac{\sin(x)}{\cos(x)}$. It has vertical asym

1. Énoncé : Résoudre dans $[\frac{\pi}{2}; \frac{3\pi}{2}]$ l'inéquation $\cos(x) < \frac{\sqrt{3}}{2}$.\n\n2. Rappel : $\cos(x)$ décroît de 0 à $\pi$ et remonte ensuite. La valeur

1. **State the problem:** We need to find the value of $d$ in the equation $y = a \sin(bx + c) + d$ that models the height $y$ of a student on a ferris wheel after $x$ seconds. 2.

1. **Problem:** In a right triangle, the opposite side to angle $\theta$ is 7 and the hypotenuse is 25. Find $\sin \theta$. 2. **Formula:** $\sin \theta = \frac{\text{opposite}}{\t

1. **State the problem:** Find the exact value of $$(\sec 30^\circ)(\cos 30^\circ)(\tan 60^\circ)(\cot 60^\circ)$$. 2. **Recall the definitions and values:**

1. **State the problem:** Find all $x$-intercepts of the function $$f(x) = 6 \sin^2 x + 3 \sin x - 3 = 0$$ on the interval $$[-\frac{\pi}{2}, \frac{\pi}{2}]$$. 2. **Rewrite the equ

1. **State the problem:** We need to find the width of the river, which is the horizontal distance between the surveyor and the pole. 2. **Identify the known values:**

1. **State the problem:** Given $\sin(A+B) = 0.75$ and $\sin(A-B) = 0.43$, find the value of $\cos A \sin B$ to the nearest hundredth. 2. **Recall the sine addition and subtraction

1. The problem states that the angle is 81.4 degrees. 2. Since no further context or question is provided, we interpret this as identifying or using the angle value.

1. **State the problem:** Find the angle $\theta$ in a right triangle where the hypotenuse is 8 m and the adjacent side to $\theta$ is 5 m.

1. **Problem statement:** Solve the equation $$\cos 3x + \cos 2x + \cos x = 0$$ for $$0 \leq x \leq 2\pi$$. 2. **Formula and identities:** Recall the cosine addition formulas and s

1. Le problème est de comprendre pourquoi $\arctan(-1,742) = -1,05$ radians. 2. La fonction $\arctan(x)$ donne l'angle dont la tangente est $x$. Elle est définie pour tout réel $x$

1. The problem is to evaluate $\csc^2(45^\circ) - \tan^2(45^\circ)$.\n\n2. Recall the trigonometric identities and values:\n- $\csc(\theta) = \frac{1}{\sin(\theta)}$\n- $\tan(\thet

1. **Problem Statement:** We need to find the length of side BC in a right triangle where angle A is 35°, side AC (adjacent to angle A) is 2 units, and the right angle is at vertex

1. **State the problem:** We need to find the height of a balloon observed from two stations X and Y, which are 3000 feet apart. Given angles are horizontal angles and angle of ele

1. **Problem statement:** A balloon is observed from two stations, X and Y, which are 300 meters apart. At station X, the horizontal angle between the balloon and point C is $75^\c

1. **Problem:** Given a right triangle \(\triangle ACB\) with side \(b=12\) and angle \(\angle A=12^\circ\), find the missing parts (sides \(a\), \(c\) and angle \(\angle B\)). 2.

1. The problem is to understand why $\cos \frac{\pi}{2} = 0$. 2. Recall that the cosine function relates to the unit circle, where $\cos \theta$ is the x-coordinate of the point on

1. مسئله: بررسی اینکه آیا نقطه $\left(\frac{\sqrt{3}}{2}, \frac{\sqrt{3}}{2}\right)$ روی نیم‌خط $OP$ در ربع مثبت محور $OX$ قرار دارد یا خیر. 2. فرمول و توضیح: نیم‌خط $OP$ معمولاً ب