📏 trigonometry

1. **State the problem:** Solve the equation $$(\tan 3\theta + \sec 3\theta)^2 = 6$$ for $$0^\circ \leq \theta \leq 180^\circ$$. 2. **Recall the identity:** We know that $$\tan x +

1. **State the problem:** Solve the equation $$\sqrt{3} \csc\left(2\theta + \frac{3\pi}{4}\right) = 2$$ for $$-\pi < \theta < \pi$$, giving answers in terms of $$\pi$$. 2. **Rewrit

1. **State the problem:** Verify the identity $$\sin^6 A + \cos^6 A + 3 \sin^2 A \cos^2 A = 1.$$\n\n2. **Recall important formulas:** We know that $$\sin^2 A + \cos^2 A = 1.$$ Also

1. The problem is to verify or simplify the expression \( \frac{\sin x}{1-\sin x} + \frac{\sin x - 1}{1 + \sin x} = 4 \sec x \tan x \). 2. Start by simplifying the left-hand side (

1. **State the problem:** Verify if the identity $\sin(a-b)\sin(a-b) = \sin^2 a - \sin^2 b$ is true. 2. **Recall the formula:** The left side is $\sin^2(a-b)$.

1. **Problem statement:** Prove the identity $$\frac{\tan x - \sin x}{2 \tan x} = \sin^2 \left(\frac{x}{2}\right)$$. 2. **Recall formulas:**

1. The problem is to find side AC in triangle ABC using the Law of Sines. 2. The Law of Sines formula is $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ where $a,b,c$ ar

1. **State the problem:** Simplify the expression $$\frac{1}{\sin x} + \frac{1}{\cos x} \over \frac{1}{\sin x} - \frac{1}{\cos x}$$. 2. **Write the expression clearly:**

1. The problem is to verify the trigonometric identity: $$\frac{1}{\sin x} + \frac{1}{\cos x} \div \left(\frac{1}{\sin x} - \frac{1}{\cos x}\right) = \frac{\cos^2 x - \sin^2 x}{1 -

1. Problem: Verify and prove the identity \( \sec x = \frac{\sin 2x}{\sin x} - \frac{\cos 2x}{\cos x} \) using \( x = \frac{\pi}{4} \). 2. Recall the double-angle formulas:

1. The problem is to find all solutions to the equation $\sin(4x) = 0$. 2. Recall that $\sin(\theta) = 0$ when $\theta = k\pi$ for any integer $k$.

1. **Problem statement:** Find the missing side $x$ in the right-angled triangle with a side of 3 cm and an angle of 33° opposite to $x$. 2. **Formula and rules:** In a right trian

1. The problem involves analyzing two trigonometric functions $f(x)$ (solid line) and $g(x)$ (dashed line) graphed against $x$. 2. From the graph description:

1. **State the problem:** We are given a sine function in the form $$y = a \sin(b(x - c)) + d$$ and a graph with specific points. We need to find the value of $$a + b + c + d$$ to

1. **State the problem:** We are given a sinusoidal function of the form $$y = a \sin\bigl(b(x - c)\bigr) + d$$ and a graph description. We want to identify the parameters $a$, $b$

1. We are given the function $$f(x) = \frac{\sin x - \cos x}{\sin x + \cos x}$$ and asked to analyze or simplify it. 2. The goal is to simplify the expression by using trigonometri

1. The problem is to simplify the expression $\sec x - 4 \csc x \cdot bc$ or evaluate it if values for $b$ and $c$ are given. 2. Recall the definitions:

1. **Problem statement:** Simplify the expression $$\frac{\sin x - \sin x \cos^2 x}{\sin^2 x}$$. 2. **Recall the Pythagorean identity:** $$\sin^2 x + \cos^2 x = 1$$.

1. **State the problem:** Find the numerical value of $\sin 240^\circ + \sin 90^\circ - \cos 30^\circ$. 2. **Recall the values of trigonometric functions:**

1. **State the problem:** Solve the equation $$\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} = \frac{5}{\cos x}$$ for $$0 < x < \pi$$, giving answers in radians to 3 significant fi

1. **State the problem:** Solve the equation $$\sin x \cos x = 5x^2 \sin x \cos x$$ for $$0 < x < \pi$$. 2. **Rewrite the equation:** The equation is $$\sin x \cos x = 5x^2 \sin x