📏 trigonometry

1. We are given the equation $$2(\sin^6 x + \cos^6 x) - 3(\sin^4 x + \cos^4 x) + 1 = 0$$ and need to solve it. 2. Recall the Pythagorean identity: $$\sin^2 x + \cos^2 x = 1$$.

1. **State the problem:** We need to verify the identity $$\frac{\sin x + \cos x}{\cos^3 x} = 1 + \tan x + \tan^2 x + \tan^3 x$$. 2. **Recall definitions and formulas:**

1. **State the problem:** Prove or verify the identity $$\sin^8 x - \cos^8 x = (\sin^2 x - \cos^2 x)(1 - 2\sin^2 x \cos^2 x)$$. 2. **Recall formulas and rules:**

1. The problem is to find the function and understand the behavior of $\sin x$. 2. The sine function is a fundamental trigonometric function defined as the ratio of the length of t

1. مسئله را بیان می‌کنیم: مقدار عبارت $$A=\frac{\sin(-50^\circ)+\sin(112^\circ)}{\cos(158^\circ)-\tan(22^\circ)}$$ را بیابید. 2. فرمول‌ها و قوانین مهم:

1. The problem is to understand how the expression $\pi - \pi \sin^2 x$ can be transformed into $\pi \sin^2 x$. 2. Start with the original expression:

1. **Problem (a): Solve $\cos 2x + \cos x = 0$.** Use the double-angle formula: $\cos 2x = 2\cos^2 x - 1$.

1. Muammo: $4\cos^3 x + 3\sin x = 3\cos x + 4\sin^3 x$ tenglamaning $[0;2\pi]$ oraliqdagi eng katta va eng kichik ildizlari yig'indisini topish va $[-\pi;0]$ oraliqda nechta ildizg

1. مسئله: در مثلث قائم‌الزاویه ABC با زاویه Ă قائمه و ضلع AC برابر ۴، اگر \( \tan \alpha = \frac{3}{4} \) باشد، مقدار سینوس زاویه \( \theta \) را بیابید. 2. فرمول‌ها و نکات مهم:

1. **State the problem:** Prove that $$\frac{\sin x}{1-\cos x} + \frac{\tan x}{1+\cos x} = \cot x + \sec x \csc x$$. 2. **Recall formulas and identities:**

1. **Énoncé du problème :** Montrer que $0 < \arccos\left(\frac{3}{4}\right) < \frac{\pi}{4}$.

1. \textbf{المشكلة:} في دائرة الوحدة، نريد إثبات أن \sin \theta = y و \cos \theta = x حيث \theta هو الزاوية المرسومة في الدائرة. 2. \textbf{تعريف دائرة الوحدة:} دائرة الوحدة هي دائ

1. **Problem Statement:** Prove that on the unit circle, $\sin \theta = y$ and $\cos \theta = x$ where $(x,y)$ is a point on the circle corresponding to angle $\theta$.

1. **Problem statement:** Lara stands 40 meters from a tree and observes the tree's top at a 30° angle of elevation and the bottom of its reflection in the water at a 45° angle of

1. **Problem:** Calculate $\cos\left(\frac{\pi}{3}\right)$. 2. **Formula and rules:** The cosine of an angle in radians can be found using the unit circle or known special angles.

1. (a) Prove the identity $\cos 3\theta \equiv 4 \cos^3 \theta - 3 \cos \theta$ by expressing $3\theta$ as $2\theta + \theta$. Use the cosine addition formula: $\cos(a+b) = \cos a

1. **Problem (a):** Prove the identity $$\cos 3\theta \equiv 4 \cos^3 \theta - 3 \cos \theta$$ by expressing $$3\theta$$ as $$2\theta + \theta$$. 2. Use the cosine addition formula

1. Solve the equation $3 \cot x - 4 \cot 2x = 3$ for $0^\circ \leq x \leq 180^\circ$. 2. (a) Express $7 \sin \theta + 24 \cos \theta$ in the form $R \cos (\theta - \alpha)$, where

1. **Problem Statement:** A kite is flying at a height of 75 m from the ground, attached to a string inclined at 60° to the horizontal. Find the length of the string to the nearest

1. **State the problem:** Convert the Cartesian coordinate $(2, -6)$ to polar coordinates $(r, \theta)$ where $0 \leq \theta < 2\pi$. 2. **Recall the formulas:**

1. The problem asks to find the x-values of all vertical asymptotes of the function $$y = \csc(5x)$$ in the interval $$[0, 2\pi)$$. 2. Recall that $$\csc(\theta) = \frac{1}{\sin(\t