📏 trigonometry

1. **State the problem:** We need to find the length $x$ of a retractable awning that is attached to a wall at a height of $y=12$ feet and lowers at an angle of $50^\circ$ from the

1. Evaluate the following using a calculator and write answers to four decimal places. **a.** $\sin(37^\circ)$

1. **Problem:** Find the six trigonometric function values of the specified angle given the triangle with angle 30° between side 18 (vertical) and side 24 (horizontal), right angle

1. The problem states: Given that $f(x) = 9$, find $f(x) = \sin x$. 2. This seems to be a misunderstanding because $f(x)$ cannot be both a constant 9 and $\sin x$ simultaneously.

1. Problem: Given point $P(-2,2)$ lies on the terminal arm of angle $\theta$ in standard position, determine which statement about $\tan\theta$, $\sin\theta$, $\cos\theta$, and $\c

1. The problem is to find the range of the variable $\theta$ given the inequality $0 < \theta \leq 360$ and the expression $\cos^2 x + \sin^2 x$. 2. Recall the Pythagorean identity

1. The problem is to understand the range of the variable $\theta$ given the inequality $0 < \theta \leq 360$. 2. This means $\theta$ is an angle measured in degrees.

1. The problem is to simplify the expression $\sin 2\theta - \sin^2 \frac{\theta}{2}$.\n\n2. Recall the double-angle identity for sine: $\sin 2\theta = 2 \sin \theta \cos \theta$.

1. **State the problem:** Solve the equation $\tan 2x + 3 \tan x = 0$ for $x$. 2. **Recall the double-angle formula for tangent:**

1. **State the problem:** Solve the equation $\tan^2 x + 3 \tan x = 0$ for $x$. 2. **Use substitution:** Let $t = \tan x$. The equation becomes:

1. **State the problem:** Solve the equation $$3\cos^3\theta + 3\cos\theta + 4\cos\theta - 4 = 0$$ for $$0^\circ < \theta \leq 270^\circ$$ and determine the number of roots. 2. **C

1. **State the problem:** Find the fundamental period of the function $$f(x) = \tan\left(\frac{2x - 1}{3}\right)$$. 2. **Recall the period of tangent function:** The basic tangent

1. **Énoncé du problème :** Résoudre l'équation $$\cos x - (\sqrt{2} - 1) \sin x = \sqrt{2} - \sqrt{2}$$ et montrer les propriétés liées à l'angle $$\alpha \in ]0, \frac{\pi}{2}[$

1. **State the problem:** Simplify the expression $\frac{\cos \theta}{1 + \tan^2 \theta}$. 2. **Recall the Pythagorean identity:** We know that $1 + \tan^2 \theta = \sec^2 \theta$.

1. **State the problem:** Solve the equation $$\tan 2x + 3 \tan x = 0$$ for $x$. 2. **Recall the double-angle formula for tangent:** $$\tan 2x = \frac{2 \tan x}{1 - \tan^2 x}$$.

1. **State the problem:** Simplify the expression $$\csc^2 x \tan x - \sin x \sec x.$$\n\n2. **Recall definitions and identities:**\n- $$\csc x = \frac{1}{\sin x}$$\n- $$\sec x = \

1. **State the problem:** Simplify the expression $$\csc^2 x \tan x - \sin x \sec x.$$\n\n2. **Recall definitions and identities:**\n- $$\csc x = \frac{1}{\sin x}$$\n- $$\sec x = \

1. **State the problem:** Simplify the expression $$\sin\left(\frac{\pi}{2} - x\right) + \cos x \tan^2 x.$$ 2. **Recall the identity:** $$\sin\left(\frac{\pi}{2} - x\right) = \cos

1. **State the problem:** Simplify the expression $$\sin\left(\frac{\pi}{2} - x\right) + \cos x \tan^2 x.$$\n\n2. **Recall trigonometric identities:**\n- $$\sin\left(\frac{\pi}{2}

1. **Problem Statement:** Solve the equation $$\sin x + \cos 2x = 0$$ for $$0 \leq x \leq \pi$$. 2. **Recall the double angle formula:** $$\cos 2x = 1 - 2\sin^2 x$$ or $$\cos 2x =

1. **Problem statement:** We have two buildings on level ground, 10 m apart. From the top of the first building, the angle of elevation to the top of the skyscraper is 70° and the