📏 trigonometry

1. Bài toán yêu cầu ta trình bày các công thức lượng giác cơ bản. 2. Công thức lượng giác giúp ta tính toán các giá trị của các hàm sin, cos, tan dựa trên các góc.

1. **Problem Statement:** We need to model the height $h(t)$ of a passenger on a Ferris wheel over time $t$ seconds. Given:

1. **State the problem:** Find all solutions to the equation $$\sin \theta + \sin 4\theta = \sin 2\theta + \sin 3\theta$$ in the interval $$-\pi < \theta \leq \pi$$ and determine t

1. **Problem Statement:** Determine the length of side $c$ in the right triangle with $\angle C = 90^\circ$, $BC = 5$ ft, $\angle A = 30^\circ$, and side $BA = c$.

1. **State the problem:** Find the value of $\arccos\left(\frac{4}{3 \times \sqrt{2}}\right)$. 2. **Recall the domain of arccos:** The function $\arccos(x)$ is defined only for $x$

1. **State the problem:** Find the value of $\arccos\left(\frac{4}{3} \times \sqrt{2}\right)$. 2. **Recall the domain of arccos:** The function $\arccos(x)$ is defined only for $x$

1. **State the problem:** Solve the trigonometric equation $3 \sin \theta - 4 \cos \theta = 2$ for $\theta$. 2. **Formula and approach:** We use the identity that any expression of

1. **State the problem:** Find the real solutions for the equations: (a) $3 \sin \theta - 4 \cos \theta = 2$,

1. **State the problem:** We need to sketch one cycle of the function $$f(x) = 3\cos 2(x - 30^\circ) - 2$$ using transformations, then find its domain and range.

1. The problem is to evaluate $\sin(4)$.\n\n2. The sine function, $\sin(x)$, gives the ratio of the opposite side to the hypotenuse in a right triangle for an angle $x$ measured in

1. **Problem:** Simplify the expression $ (1 - \cos\theta)(1 + \cos\theta) $. 2. **Formula:** Use the difference of squares identity: $ (a - b)(a + b) = a^2 - b^2 $.

1. **Énoncé du problème :** Calculer l'expression $$\cos^2(45^\circ) - \sin^2(-270^\circ) \times \tan^2(240^\circ)$$ sans calculatrice. 2. **Rappel des formules et règles important

1. Énonçons le problème : On a l'équation $8 \tan x = 3 \cos x$ avec $0^\circ < x < 180^\circ$. Il faut déterminer la valeur de $\sin x$. 2. Rappelons que $\tan x = \frac{\sin x}{\

1. **State the problem:** A 5.5-foot tall man is standing with the sun shining at an angle of depression of 55°. We need to find the length of his shadow.

1. **Énoncé du problème :** Philippe veut fixer un panier de basket à une hauteur réglementaire de 3,05 m.

1. The problem asks to evaluate the expression a) $\tan 2x$ for $\pi \leq x \leq \frac{3\pi}{2}$ given $\cos x = \frac{2}{45}$.\n\n2. Recall the double angle formula for tangent: $

1. **State the problem:** We want to write an equation equivalent to \(y = -6\cos\left(x + \frac{\pi}{2}\right) + 4\) using the sine function. 2. **Recall the trigonometric identit

1. **State the problem:** Prove the trigonometric identity $$\sec^4 x - 1 = 2 \tan^2 x + \tan^4 x.$$\n\n2. **Recall the fundamental identity:** $$\sec^2 x = 1 + \tan^2 x.$$ This is

1. **State the problem:** Susan is at the top of a 2.1 m high wall at the edge of an 8 m wide beach. She measures the angle of depression to a swimmer in the sea as 6°. We need to

1. **State the problem:** Solve the equation $\left(c \sin x + \cos x\right)^2 = 2$ for $x$. 2. **Recall the formula and rules:** The square of a sum is expanded as $\left(a+b\righ

1. Énonçons le problème : Résoudre sur l'intervalle $]-\pi; \pi]$ le système d'inéquations : $$\cos(x) \leq -\frac{\sqrt{2}}{2}$$