📏 trigonometry

1. The problem is to find the value of $\cot(x + \frac{\pi}{3})$ or express it in terms of $x$. 2. Recall the cotangent addition formula:

1. The problem is to simplify or understand the expression $\tan(x-\frac{\pi}{6})$. 2. We use the tangent subtraction formula:

1. **State the problem:** We want to simplify or express the trigonometric expression $\cos(x + \frac{\pi}{3})$. 2. **Formula used:** Use the cosine addition formula:

1. **State the problem:** Prove that $$\frac{1}{\sin A} = \frac{\cos A (1 + A)}{\tan(\tan A)}$$. 2. **Analyze the given expression:** The left side is $$\frac{1}{\sin A}$$, which i

1. **Problem statement:** Given that $\cos \theta = \frac{1}{2}$ and $\sin \theta = \frac{\sqrt{3}}{2}$, find the measure of angle $\theta$. 2. **Recall the unit circle values:** T

1. **Problem Statement:** Given that $\sin \theta = -1$ and $\cos \theta = 0$, find the measure of the angle $\theta$. 2. **Recall the unit circle values:**

1. **State the problem:** Prove that $$\arctan x + \arctan \frac{1}{x} = \frac{\pi}{2}$$ for $x > 0$. 2. **Recall the formula for the tangent of a sum:**

1. **State the problem:** Prove that $$\arccos\sqrt{\frac{2}{3}} - \arccos\frac{\sqrt{6}+1}{2\sqrt{3}} = \frac{\pi}{6}$$. 2. **Recall the formula:** For any angles $A$ and $B$, the

1. **State the problem:** We want to prove that $$\arccos\frac{5}{13} = 2\arctan\frac{2}{3}$$. 2. **Recall the formulas:**

1. **State the problem:** We want to prove the identity $$\frac{\pi}{4} = \arctan\frac{1}{3} + \arctan\frac{1}{7} + \arctan\frac{1}{13} + \arctan\frac{1}{21} + \cdots$$ 2. **Recall

1. **State the problem:** Prove that $$\arctan\left(\frac{1}{n^2+n+1}\right) = \arctan\left(\frac{1}{n}\right) - \arctan(x)$$ for some value of $x$. 2. **Recall the arctan subtract

1. **Problem statement:** We have four statues A, B, C, and D with given angles and lengths. We need to find angles ACD, BCD, ACB, and length AB.

1. **Problem Statement:** Evaluate the exact values of various trigonometric expressions and convert Cartesian coordinates to polar form, determine signs of trig ratios, and solve

1. **State the problems:** We have two equations to solve for $x$:

1. **Problem Statement:** Find the minimum value of $$2 \cos \theta + \frac{1}{\sin \theta} + \sqrt{2} \tan \theta$$ where $$\theta$$ is an acute angle. 2. **Understanding the prob

1. **Problem 1: Find the minimum value of** $2 \cos \theta + \frac{1}{\sin \theta} + \sqrt{2} \tan \theta$, where $\theta$ is an acute angle. 2. **Using the Arithmetic Mean - Geome

1. **State the problem:** Solve the trigonometric equation $$2\cos(2x) - \cos(x) - 1 = 0$$. 2. **Use the double-angle identity:** Recall that $$\cos(2x) = 2\cos^2(x) - 1$$.

1. **Problem statement:** Express $6 \sin^2 \theta \cot 2\theta + 4 \sin \theta \cos \theta$ in terms of $\sin 2\theta$ and $\cos 2\theta$ only.

1. **State the problem:** Solve the trigonometric equation $$\cos 6x + \sin 6x = 1 - 3 \sin^2 x \cos^2 x.$$\n\n2. **Recall relevant formulas and identities:**\n- Use the double-ang

1. **Problem statement:** Given $\tan \theta = 2$, find $\tan(3\theta)$.\n\n2. **Formula used:** The triple-angle formula for tangent is:\n$$\tan(3\theta) = \frac{3\tan\theta - \ta

1. نبدأ ببيان المشكلة: لدينا المعادلة $\sin \theta = \cos \theta$ حيث $\theta$ في الفترة $[0, 2\pi]$. 2. نستخدم العلاقة الأساسية بين الجيب وجيب التمام: \( \sin \theta = \cos \theta