📏 trigonometry

1. **State the problem:** We want to find the formula for a trigonometric function of the form $$h(x) = a \cos(bx + c) + d$$

1. We are given a vertical pole casting a shadow of 1.8 meters when the sun's elevation angle is 63.5 degrees. We need to find the height of the pole. 2. The problem can be solved

1. **Problem Statement:** Two signposts A and B are 11 km apart on a straight highway. An airplane is flying above the highway. The angles of depression from the airplane to A and

1. The problem is to understand the terms 'cahsign' which likely refers to 'cosine, adjacent, hypotenuse sign,' a mnemonic for trigonometric ratios. 2. The cosine of an angle in a

1. The problem is to find the value of $ (1 + \tan A)(1 + \tan B)$ where $A = 40^\circ$ and $B = 5^\circ$. 2. Substitute the given values:

1. The problem involves two expressions:\n$$\frac{1}{1} + \frac{1}{\text{something}} = 1$$\nand\n$$\frac{1 + \sin^2 A}{1 + \csc^2 A}$$\nWe need to analyze and simplify these.\n\n2.

1. **Problem statement:** Given $\sin \alpha = \frac{3}{5}$ with $\alpha$ in the first quadrant, find: - 5.3.1 $\tan \alpha$

1. Problem 5.1: Simplify $$\sin(90^\circ - x) \cdot \cos(180^\circ + x) + \tan x \cdot \cos x \cdot \sin(x - 180^\circ)$$ Step 1: Use angle identities:

1. The problem states that Pushkar observes the top of a pole $23^3$ m high with an angle of elevation of $30^\circ$, and the distance between Pushkar and the pole is 66 m. 2. Firs

1. **State the problem:** Pushkar observes the top of a pole that is $23^3$ meters high. The angle of elevation from Pushkar to the top of the pole is $30^\circ$, and the horizonta

1. დავიწყოთ პირველით: თუ $\sin x = \frac{\sqrt{2}}{2}$ და $x\in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, მაშინ $x$ არის ის კუთხე, რომლის სინი არის $\frac{\sqrt{2}}{2}$. პოპულარ

1. Given $\sin x = \frac{\sqrt{2}}{2}$ and $x \in [-\frac{\pi}{2}, \frac{\pi}{2}]$, recognize that $\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$. Since $\frac{\pi}{4}$ is within the in

1. **State the problem:** Show that $$\tan\left(\frac{45^\circ + \theta}{2}\right) \cdot \tan\left(\frac{45^\circ - \theta}{2}\right) = \frac{\sqrt{2} \cos \theta - 1}{\sqrt{2} \co

1. **Resolve a equação trigonométrica** $5\sin x - 6\cos x - 5 = 0$.\n Podemos reorganizar para isolar as funções trigonométricas:\n

1. State the problem: Express $3\cos \theta - 4\sin \theta$ in the form $R\cos(\theta - \alpha)$ and solve the equation $3\cos \theta - 4\sin \theta = 0$ for $0 \leq \theta \leq 2\

1. Let's start by understanding the quadrants. The first quadrant is where angles measure from $0^\circ$ to $90^\circ$, and the fourth quadrant is where angles measure from $270^\c

1. We are given that \(\tan(x) = \csc(x) - \sin(x)\) and need to prove that \(\tan(x^2)\left(\frac{x}{2}\right) = -2 + \sqrt{5}\).\n\n2. Start by expressing \(\csc(x)\) in terms of

1. The problem asks why the cosine function is positive in the first and fourth quadrants of the unit circle. 2. Recall that cosine of an angle $\theta$ in the unit circle is the $

1. **State the problem:** We need to find all angles $x$ between 0 and 360 degrees such that $$\cos(2x) = \frac{\sqrt{3}}{2}.$$

1. The problem asks for three common applications of trigonometric identities and how to create steps to prove these identities using a flowchart. 2. Three common applications of t

1. **State the problem:** Prove that if $A = \frac{\pi}{12}$, then $$\tan A \cdot \tan 3A \cdot \tan 5A \cdot \tan 7A \cdot \tan 11A = 1.$$\n\n2. **Substitute $A = \frac{\pi}{12}$: