📏 trigonometry

1. **State the problem:** Simplify the expression $$\frac{1-\sin^4 x}{\cos^4 x}$$. 2. **Recall the formula and identities:** Use the difference of squares formula: $$a^2 - b^2 = (a

1. The problem is to find the value of $\theta$ given some context or equation involving $\theta$. 2. To solve for $\theta$, we need an equation or expression where $\theta$ is the

1. The problem is to find the value of $\cos\theta$ for a given angle $\theta$. 2. The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent

1. Problem: Find the value of $\theta$ if $\tan \theta = 1$. The tangent function equals 1 at angles where the opposite and adjacent sides are equal. The principal angle is $\theta

1. **Evaluate $\tan(\cos(-\frac{11\pi}{2}))$** - First, find $\cos(-\frac{11\pi}{2})$.

1. The problem is to evaluate $\cos(\cos 270^\circ)$.\n\n2. First, recall that $\cos 270^\circ$ means the cosine of 270 degrees.\n\n3. Using the unit circle, $\cos 270^\circ = 0$.\

1. **State the problem:** Find the exact value of $\csc\left(-\frac{5\pi}{6}\right)$.\n\n2. **Recall the definition:** $\csc(\theta) = \frac{1}{\sin(\theta)}$. So we need to find $

1. **State the problem:** Evaluate $\cot 300^\circ$. 2. **Recall the definition:** $\cot \theta = \frac{\cos \theta}{\sin \theta}$.

1. The problem is to evaluate $\tan 240^\circ$. 2. Recall that the tangent function is periodic with period $180^\circ$, and $\tan(\theta) = \frac{\sin \theta}{\cos \theta}$.

1. The problem asks: In which quadrants is the sine function negative? 2. Recall that the sine function, $\sin(\theta)$, represents the y-coordinate of a point on the unit circle a

1. **State the problem:** Simplify and verify the identity $$\frac{\tan \alpha}{\sec \alpha - 1} = \frac{\sec \alpha + 1}{\tan \alpha}$$. 2. **Recall basic trigonometric identities

1. **State the problem:** Prove the trigonometric identity $$\sin A + \sin B = 2 \sin \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right)$$. 2. **Recall the sum-to-produ

1. **State the problem:** Simplify the expression $$\frac{(\sin x)^3 - (\cos x)^3}{\sin x + \cos x} = \frac{(\csc x)^2 - \cot x + 2(\cos x)^2}{1 - (\cot x)^2}$$. 2. **Recall formul

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