📏 trigonometry

1. Vamos resolver a equação $4 \cos^2(x) \sin(x) - \sin(x) = 0$. 2. Primeiro, fatoramos $\sin(x)$ em evidência:

1. **Problem Statement:** (a) Given $f(\theta) = 5 \cos \theta + \sin \theta$, express it in the form $f(\theta) = R \cos(\theta - \alpha)$, with $R>0$ and $0 \leq \alpha \leq \fra

1. Given \(\cos \theta = -\frac{1}{2}\) and \(\pi < \theta < \frac{3\pi}{2}\), find \(\sin \theta\), \(\tan \theta\), and \(\cot \theta\). Step 1: Identify the quadrant.

1. The problem is to solve the equation $\sin\left(\frac{3}{4} - 4x\right) = -\frac{1}{2}$.\n\n2. Recall that $\sin\theta = -\frac{1}{2}$ at angles $\theta = \frac{7\pi}{6} + 2k\pi

1. **State the problem:** Solve the equation $$\sin\left(\frac{3}{4} - 4x\right) = -\frac{1}{2}.$$\n\n2. **Recall the solutions for sine:** The sine function equals $$-\frac{1}{2}$

1. **Problem:** Simplify the expression $\sin^4\theta + \cos^4\theta$ and find its equivalent form. 2. Start by recognizing the expression is a sum of fourth powers of sine and cos

1. Let's clarify the problem: it seems you're referring to an expression or function where the power applies only to the sine term. 2. For example, if you have $y = \sin^n x$, this

1. Problem 5: A pole 6 m high casts a shadow 2.3 m long. Find the Sun's elevation angle. - The elevation angle $\theta$ satisfies $\tan \theta = \frac{\text{height}}{\text{shadow l

1. The problem asks us to find the value of $\tan \frac{\pi}{4}$.\n\n2. Recall that $\frac{\pi}{4}$ radians is equivalent to 45 degrees.\n\n3. The tangent of 45 degrees (or $\frac{

1. The problem states the function as $y=4\sin\Theta$. 2. This is a trigonometric function where the amplitude is 4 because it is the coefficient of $\sin\Theta$.

1. The question asks if I am knowledgeable about trigonometry, specifically the sine rule. 2. The sine rule states that in any triangle, the ratio of the length of a side to the si

1. Given the equation $$\sec\theta \cot\theta = \csc\theta$$, we want to verify or simplify it. Recall the definitions:

1. Let's start by understanding what trigonometric ratios are. 2. Trigonometric ratios are relationships between the lengths of the sides of a right triangle.

1. The problem is to understand the trigonometric function $\sin \theta$. 2. $\sin \theta$ represents the ratio of the length of the opposite side to the hypotenuse in a right tria

1. **Problem:** Given \(\sin A = \frac{4}{5}\) where \(A\) is in quadrant 3, and \(\cos B = -\frac{1}{5}\) where \(B\) is in quadrant 2, find \(\cot (A - B)\). 2. Find \(\cos A\) k

1. Let us start by stating the problem: Given the equation $$(a)(1+m) \sin(\theta + \alpha) = (1-m) \cos(\theta - \alpha)$$

1. The problem states the inequality: $$4 \tan t + 5 \sin \theta \geq 20.$$ 2. To understand this inequality, note that $$\tan t$$ and $$\sin \theta$$ are trigonometric functions w

1. Let's start with the basics of trigonometry: Trigonometry studies the relationships between the angles and sides of triangles, especially right triangles. 2. The primary functio

1. **State the problem:** We have a right triangle formed by points O (observer), R (base of tower), and T (top of tower).

1. **State the problem:** Prove the identity \( \tan A - \cot A = 2 \tan(2A) \). 2. **Rewrite cotangent:** Recall that \( \cot A = \frac{1}{\tan A} \).

1. **State the problem:** We have a right triangle formed by the observer's position $O$, the base of the tower $R$, and the top of the tower $T$. Given: $OR=84$ m, initial angle o