📏 trigonometry

1. **State the problem:** Given $\sin A = \frac{8}{17}$$ with $A$ obtuse, and $\cos B = \frac{5}{13}$ with $B$ acute, find $\sin(2A + B)$ without using tables or direct calculation

1. Vamos preencher a tabela com os valores dos senos, cossenos e tangentes para os ângulos dados: 0°, 30°, 45°, 60° e 90°. 2. Fórmulas e valores importantes:

1. **Problem Statement:** We need to sketch the graph of the function $f(x) = 1 + \cos x$.

1. **State the problem:** We need to analyze and sketch the graph of the function $$f(x) = 1 + \cos x$$. 2. **Recall the cosine function properties:** The cosine function $$\cos x$

1. The problem asks to sketch the graph of the function $f(x) = \sec x$. 2. Recall that $\sec x = \frac{1}{\cos x}$.

1. **Problem statement:** Find $\sec \theta$ for the angle $\alpha$ in the right triangle with sides $RP=18$, $PQ=27$, and hypotenuse $RQ$. 2. **Recall the definition:** $\sec \the

1. **Problem statement:** Find the hypotenuse $RQ$ of right triangle $QRP$ with right angle at $P$, sides $RP=18$ and $QP=27$. 2. **Formula:** Use the Pythagorean theorem: $$RQ^2 =

1. **Stating the problem:** Simplify the expression $$\sin 2t + \sin 5t - \cot \theta - 23 + \frac{1}{2} \sin 2t + \cos \alpha$$ where $\cot$ is assumed to be $\cot \theta$ for som

1. **State the problem:** Find the exact value of $\tan\left(-\frac{5\pi}{12}\right)$ using compound angle formulas. 2. **Recall the formula:** The tangent of a sum or difference o

1. The problem is to simplify or solve the expression $1 - 2 \sin^2(2x)$. 2. Recall the double-angle identity for cosine: $$\cos(2\theta) = 1 - 2\sin^2(\theta)$$. This means that $

1. **State the problem:** Prove the trigonometric identity $$\cos 4x = 8\sin^4 x - 8\sin^2 x + 1$$. 2. **Recall the double-angle and power-reduction formulas:**

1. **State the problem:** Explain the meaning of the function $\arctan$. 2. **Definition:** The function $\arctan$ is the inverse of the tangent function $\tan$.

1. **State the problem:** We have a right triangle with angle $A = 50^\circ$, the side adjacent to angle $A$ is 16, and we need to find the side opposite angle $A$ (denoted $a$) an

1. **State the problem:** We have a right triangle with one leg of length 23, an angle of 47°, and we need to find the other leg $a$, the hypotenuse $c$, and the remaining angle $B

1. Simplify expression 7.1: $$\frac{\sin 210^\circ \cos 300^\circ \tan 240^\circ}{\cos 120^\circ \tan 150^\circ \sin 330^\circ}$$ Recall the exact values:

1. **Problem statement:** Given point $P(-3, y)$ lies on a circle centered at the origin $O$ with radius $OP = 5$ units, and angle $\alpha = \angle XÔP$. 2. **Find $y$:** Use the d

1. **State the problem:** We have two trigonometric functions: $f(x) = a \cos x$ and $g(x) = \sin bx$ defined on $x \in [-180^\circ, 180^\circ]$. We need to find:

1. Problem 9: Simplify $\sin\left(\frac{3\pi}{2} + \alpha\right) \cot\left(\pi + \beta\right)$. Formula: Use angle addition formulas and periodicity of trig functions.

1. **Problem statement:** Given $\sin \theta = \frac{\sqrt{21}}{5}$ and $\frac{\pi}{2} \leq \theta \leq \pi$, find $\sin 2\theta$ and $\cos 2\theta$ without using a calculator. 2.

1. The problem asks to find the amplitude of the sinusoidal function shown in the graph. 2. The amplitude of a sinusoidal function $f(x) = A \sin(Bx + C) + D$ or $f(x) = A \cos(Bx

1. **Problem Statement:** Find the period of the sinusoidal function graphed, which oscillates between -5 and 1 on the y-axis and completes one full cycle between consecutive multi