📏 trigonometry

1. **Problem 6:** Use the Law of Cosines to find angle $\theta$ at vertex B in a triangle with sides $AB=42.75$, $BC=64.01$, and $AC=35.51$. 2. The Law of Cosines formula for angle

1. **Problem Statement:** Given the trigonometric values for angles A, B, C, D, E, and F, find the related angles in degrees and determine the quadrants where these angles lie. Als

1. **State the problem:** We need to find the exact value of $\cos(\alpha + \beta)$ given that $\tan \alpha = \frac{12}{5}$ with $\alpha$ in quadrant III, and $\cos \beta = \frac{1

1. **State the problem:** Given \( \cos \alpha = -\frac{12}{13} \) with \( \alpha \) in quadrant II, and \( \sin \beta = -\frac{8}{17} \) with \( \beta \) in quadrant III, find \(

1. **State the problem:** We need to find $\tan(\alpha + \beta)$ given $\tan \alpha = \frac{3}{4}$ with $\alpha$ in quadrant I, and $\cos \beta = \frac{4}{5}$ with $\beta$ in quadr

1. **State the problem:** We need to find $\sin(\alpha - \beta)$ given $\cos \alpha = \frac{12}{13}$ with $\alpha$ in quadrant IV, and $\tan \beta = \frac{5}{12}$ with $\beta$ in q

1. **Problem Statement:** Given the functions $f(x) = \cos 2x$ and $g(x) = \tan x + 2$ for $x \in [-180^\circ, 90^\circ]$, we need to sketch both graphs on the same axes, label int

1. **Problem Statement:** Given the functions $f(x) = \cos 2x$ and $g(x) = \tan x + 2$ for $x \in [-180^\circ, 90^\circ]$, we need to:

1. **Verify** $\csc \theta \sec \theta = \tan \theta + \cot \theta$. - Recall definitions: $\csc \theta = \frac{1}{\sin \theta}$, $\sec \theta = \frac{1}{\cos \theta}$, $\tan \thet

1. **Problem:** Verify the identity $$\csc \theta \sec \theta = \tan \theta + \cot \theta$$ - Recall definitions: $$\csc \theta = \frac{1}{\sin \theta}, \quad \sec \theta = \frac{1

1. **State the problem:** Solve the equation $$2 \cos^2 x = 1$$ on the interval $$[0, 2\pi)$$. 2. **Rewrite the equation:** Divide both sides by 2 to isolate $$\cos^2 x$$:

1. **Problem Statement:** You are at the top of a watchtower 100 feet above sea level. The angle of depression to a ship in the water is 25 degrees. You need to find the distance $

1. **Problem Statement:** Find the angle of elevation $\theta$ of the top of a flagpole that is $10\sqrt{3}$ meters tall from a point on the ground $30$ meters away from the base o

1. **Problem Statement:** You are standing 70 meters from the base of a building. The angle of elevation to the top of the building is 60 degrees, and your eye level is 1.7 meters

1. **State the problem:** You are standing 70 meters from the base of a building. The angle of elevation to the top of the building is 60 degrees, and your eye level is 1.7 meters

1. **State the problem:** You are standing 70 meters from the base of a building, looking up at the top with an angle of elevation of 60 degrees. Your eye level is 1.7 meters above

1. **State the problem:** You are 70 meters from the base of a building and looking up at a 60° angle of elevation to the top. Your eye level is 1.7 meters above the ground. We nee

1. The problem asks to identify the equivalent angle to $\frac{7\pi}{6}$ radians from the given options. 2. Recall that angles in radians can be simplified or compared by subtracti

1. **Stating the problem:** We are given that $\cos(45^\circ) = \sin(\theta)$ and need to find the value of $\theta$. 2. **Recall the identity:** $\sin(\theta) = \cos(90^\circ - \t

1. The problem asks to find the value of $\cot(30^\circ)$ given that $\tan(30^\circ) = \frac{\sqrt{3}}{3}$.\n\n2. Recall the identity relating cotangent and tangent: $$\cot(\theta)

1. **Problem Statement:** Given that $\tan(\theta) = 1$, find the value of $\theta$. 2. **Formula and Explanation:** The tangent function is defined as $\tan(\theta) = \frac{\sin(\