📏 trigonometry

1. **Problem 24:** Find the longest side in triangle XYZ with angles 40° at X, 60° at Z, and side XZ = 12.0 cm. 2. **Recall:** The longest side is opposite the largest angle.

1. The problem asks whether the primary trigonometric ratios can be used in non-right triangles. 2. The primary trigonometric ratios (sine, cosine, tangent) are defined based on ri

1. Problem 19 asks for the best trigonometric tool to solve for angle $\theta$ in a right triangle with sides opposite $\theta = 8$, adjacent to $\theta = 11$, and hypotenuse $= 12

1. The problem asks to identify which formula set corresponds to a given set of trigonometric formulas. 2. The formulas given are:

1. The problem asks to identify which law or theorem is described by the formulas: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \quad \text{and} \quad \frac{\sin A}{a}

1. **State the problem:** Solve the equation $$\tan^3(x) + 3\tan(x) = 0$$ for $x$. 2. **Rewrite the equation:** Factor out $\tan(x)$:

1. **Problem statement:** We have a right-angled triangle with an angle of 20° and the adjacent side to this angle measuring 9 cm. We need to find the length of the side opposite t

1. Énonçons le problème : On a la fonction $g(x) = \arcsin\left(\frac{2x}{1+x^2}\right)$ et on cherche une autre expression équivalente de $g(x)$. 2. Rappelons une identité trigono

1. **Problem statement:** Given that $\sin \alpha \neq \frac{1}{2}$ and $\cos \beta = \frac{1}{2}$, find the value of $\alpha + \beta$. 2. **Recall the values of cosine:** $\cos \b

1. **Problem statement:** Given that $\sin \alpha \neq \frac{1}{2}$ and $\cos \beta = \frac{1}{2}$, find the value of $\alpha + \beta$. 2. **Recall the values of cosine:**

1. **State the problem:** We need to find values of $a$, $b$, and $c$ for the sinusoidal function $$y = a \sin(x + b)^\circ + c$$ given the graph's behavior over $0 \leq x \leq 360

1. The problem is to find the values of cosec, sec, and cot for the angles $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, and $90^\circ$. 2. Recall the definitions:

1. The problem asks for the period of the function $y = 2 \cos\left(\frac{x}{3}\right) - 1$. 2. Recall the general form of a cosine function: $y = A \cos(Bx - C) + D$.

1. **State the problem:** We have a curve defined by the equation $$y = a + b \sin(cx)$$ with amplitude 4 and period $$\frac{\pi}{3}$$. The curve passes through the point $$\left(\

1. **Problem Statement:** Write the equations of the sinusoidal functions for the given graph.

1. **State the problem:** Graph the function $$y = 3 \cos(4x + \pi)$$ over one period and find key coordinates. 2. **Identify the period:** The general form is $$y = A \cos(Bx + C)

1. **State the problem:** Graph the function $$y = \frac{3}{2} \sin \left( 2 \left( x + \frac{\pi}{4} \right) \right)$$ over one period and find key coordinates. 2. **Recall the fo

1. **State the problem:** Graph the function $$y = -3 + 2 \sin x$$ over a two-period interval. 2. **Recall the formula and properties:**

1. **State the problem:** Graph the function $$y = 3 \cos(4x + \pi)$$ over one period and find key coordinates. 2. **Identify the formula and period:** The general cosine function

1. **State the problem:** We need to find the original height $x$ of a leaning tree that forms a right triangle with the ground. 2. **Given:** The hypotenuse (length of the tree) i

1. **State the problem:** Solve the equation $$2\cot^2\theta + 3\csc^2\theta = 4\cot\theta + 3$$ for $$-180^\circ \leq \theta \leq 180^\circ$$. 2. **Recall identities:** We know th