📏 trigonometry

1. **State the problem:** Solve the trigonometric equation $$2\cos(x) - 1 = 0$$ for $$x$$ in the interval $$[0, 2\pi)$$. 2. **Use the formula and rules:** To solve for $$x$$, isola

1. **State the problem:** Solve the equation $$4(\sin x)^2 - 2 = 0$$ on the interval $$[0, 2\pi)$$. 2. **Rewrite the equation:**

1. **Problem:** Simplify the expression $\sqrt{1 + \cos \beta} \times \sqrt{1 - \cos \beta}$. 2. **Formula and rules:** Recall the identity for the difference of squares: $$a^2 - b

1. **Simplify** $\csc^2 x \tan^2 x \cos x$. Recall that $\csc x = \frac{1}{\sin x}$ and $\tan x = \frac{\sin x}{\cos x}$.

1. **State the problem:** Simplify the expression $\sin(x + y) - \sin(x - y)$. 2. **Recall the sine subtraction formula:** For any angles $A$ and $B$,

1. **State the problem:** Simplify the expression $$\frac{\sin(16k) + \sin(8k)}{\cos(16k) + \cos(8k)}$$. 2. **Recall sum-to-product formulas:**

1. **Problem statement:** A ladder of length 15 m leans against a vertical wall making an angle of 35° with the vertical wall. We need to find the height of the ladder above the gr

1. **Planteamiento del problema:** Tenemos dos ciudades, Samaná y Nagua, separadas por 40 km. Un helicóptero está volando entre ellas y desde cada ciudad se mide un ángulo de eleva

1. The problem asks to find the quadrant of the angle \(\theta\) based on the signs of sine, cosine, tangent, secant, and cosecant functions. 2. Recall the quadrant rules for trigo

1. The problem is to prove or verify the trigonometric identity: $$\sin(\pi - x) = \sin x$$. 2. The formula used here is the sine subtraction identity and the properties of sine fu

1. State the problem (Question 17): Given the terminal point for angle $\theta$ is $(-4,-7)$, find the values of the six trigonometric functions: $\sin\theta,\cos\theta,\tan\theta,

1. **Problem statement:** Calculate the unknown angle $x$ in triangle $PQR$ where the bearing of $Q$ from $P$ is $132^\circ$, angle $PQR$ is $56^\circ$, and the angle between north

1. **State the problem:** Given the opposite side length $20$ and an angle of $59^\circ$, find the length of the adjacent side $q$ in a right triangle. 2. **Formula used:** In a ri

1. **State the problem:** We need to find an equation for the trigonometric function $f(x)$ based on the given sinusoidal graph. 2. **Analyze the graph:** The wave oscillates betwe

1. The problem involves analyzing the given trigonometric functions to identify their midline, amplitude, period, and phase shift. 2. Recall the general form of sine and cosine fun

1. **State the problem:** We are given the function $f(x) = (1 + \tan^2 x) \tan^2 x$ and need to simplify it. 2. **Recall the trigonometric identity:** One important identity is $1

1. **State the problem:** Find the exact value of $\sin(105^\circ)$. 2. **Use the sum formula for sine:** The angle $105^\circ$ can be expressed as $60^\circ + 45^\circ$. The sine

1. **State the problem:** We want to express the equation $$3 \cos x + 2\sqrt{10} \sin x = R \cos(x - \alpha)$$ and find the value of $R$ where $R > 0$. 2. **Formula used:** The ex

1. **State the problem:** Simplify and solve the equation $$\frac{\sin\left(\frac{\pi}{2} - x\right) - 1}{1 - \cos(-x)} = \cos^2 x$$. 2. **Recall trigonometric identities:**

1. **Problem statement:** We have a right triangle ABC with a right angle at C. The angle $\theta$ is at vertex A. Side BC is opposite $\theta$, and side AC is adjacent to $\theta$

1. Énonçons le problème : Trouver les valeurs exactes de $\cos(x)$ et $\sin(x)$ telles que $$3\cos^2(x) + 5\sin(x) = 1.$$\n\n2. Rappelons la relation fondamentale entre sinus et co