📏 trigonometry

1. **State the problem:** Solve the equation $$\sec^2\theta = 5(\tan\theta - 1)$$ for $$0^\circ \leq \theta \leq 360^\circ$$. 2. **Recall the identity:** $$\sec^2\theta = 1 + \tan^

1. **State the problem:** Prove the trigonometric identity $$\frac{\cos x}{1-\sin x} - \tan x = \sec x$$. 2. **Recall definitions and formulas:**

1. **State the problem:** Prove the trigonometric identity $$\frac{\cos x}{1-\sin x} = \tan x$$. 2. **Recall the formulas and rules:**

1. Problem: Find the exact values without a calculator for: a) $\cos 40^\circ \cos 50^\circ - \sin 40^\circ \sin 50^\circ$

1. **Problem statement:** (a) Find the exact value of $\tan 30^\circ$.

1. **State the problem:** We are given values of $x$ and $y$ that satisfy the equation $$y = a \sin x^\circ + b$$ and need to find $y$ when $x = 45^\circ$. 2. **Use the given data

1. The problem is to find the exact value of $\sin 60^\circ$. 2. Recall that $60^\circ$ is an angle in an equilateral triangle where all sides are equal and all angles are $60^\cir

1. The problem is to find the value of $\cos 30^\circ$. 2. Recall the cosine function in a right triangle is defined as the ratio of the length of the adjacent side to the hypotenu

1. The problem asks to find the exact value of $\tan 30^\circ \times \tan 30^\circ$ in simplest form. 2. Recall the exact value of $\tan 30^\circ$ is $\frac{1}{\sqrt{3}}$.

1. The problem is to understand the symbol $\theta$, which is commonly used in mathematics and physics to represent an angle. 2. $\theta$ is a variable often used in trigonometry,

1. Stating the problem: Simplify and verify the trigonometric identity $$\cos \theta \sqrt{\cot^2 \theta + 1} = \sqrt{\csc^2 \theta - 1}$$. 2. Recall the Pythagorean identities:

1. **Problem:** Prove the identity \(\sin 3x = 3 \sin x - 4 \sin^3 x\). 2. **Formula and rules:**

1. **Problem Statement:** We are asked to sketch and analyze the graphs of sine and cosine functions with horizontal shifts (phase shifts), amplitude changes, and combined transfor

1. **Problem:** Find the general solution of the equation $\cot \theta = 0$. 2. **Formula and rules:** Recall that $\cot \theta = \frac{\cos \theta}{\sin \theta}$. The cotangent fu

1. **State the problem:** Solve the equation $$\cos 2x + 3 \sin x = 2$$ for $$0^\circ \leq x \leq 90^\circ$$. 2. **Recall the double-angle identity:** $$\cos 2x = 1 - 2 \sin^2 x$$.

1. **Stating the problem:** We have a bridge across a valley that is 150 m long. The valley walls make angles of 60° and 54° with the bridge. We need to find the depth of the valle

1. **Problem Statement:** You have measurements for the reddish orange triangle and want to find the measurements for the white and blue triangles in a composite figure made of rig

1. The problem asks to determine the length of the period of a periodic sinusoidal function given its graph. 2. The period of a sinusoidal function (sine or cosine) is the horizont

1. **State the problem:** We need to determine the length of the period of a sinusoidal function that oscillates between 5 and -5. 2. **Given information:** The function peaks at a

1. The problem is to find how many solutions satisfy the equation $\sin x = x$ in the interval $[0, \pi]$. 2. We use the fact that $\sin x$ is a trigonometric function oscillating

1. **Problem:** Simplify the function $$f(x) = \frac{\sin x}{1 - \cos x}$$ for $$x \neq 2\pi k, k \in \mathbb{Z}$$. 2. **Formula and identities used:**