📏 trigonometry

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

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

1. **State the problem:** Ravish measures the angle of elevation to the roof of a building and to the center of a logo on the building from point P, which is 24 m from the building

1. **Stating the problem:** We have a right triangle representing a building and the point of observation P. The base adjacent to angle $\theta$ is 9 m, and we want to find $\tan \

1. Planteamos el problema: Desde un punto en el terreno se observa una torre con un ángulo de elevación $\alpha$. Desde la mitad de la distancia, el ángulo de elevación es el compl

1. **State the problem:** Find all values of $A$ such that $\tan A = -\sqrt{3}$ with $0 \leq A \leq 720^\circ$. 2. **Recall the tangent values:** The tangent of an angle is $-\sqrt

1. The problem is to create tools to study trigonometry. 2. Trigonometry studies relationships between angles and sides in triangles, especially right triangles.

1. **Stating the problem:** We have two functions, $y=\cos x$ and $y=\sin x$, graphed on the Cartesian plane. The curves intersect at points that create three shaded regions labele

1. The problem is to simplify or understand the expression $y = \sqrt{\sin^4(x) + \cos(x)}$. 2. Recall the Pythagorean identity: $\sin^2(x) + \cos^2(x) = 1$. However, here we have

1. The problem is to simplify the expression $\sin(x + \frac{\pi}{2})$. 2. We use the sine addition formula: $$\sin(a + b) = \sin a \cos b + \cos a \sin b$$

1. **State the problem:** Graph one cycle of the function $$y = 3 \cos\left(x + \frac{\pi}{2}\right)$$ and explain its characteristics. 2. **Formula and rules:** The general cosine

1. **Problem statement:** We have triangle BFG with sides BF = 36 km, FG = 55 km, and angles at F and B given. We need to find:

1. **State the problem:** We have a cliff 100m high with a lighthouse 30m tall on top. A boat sees the top of the lighthouse at an angle of elevation of 20°. 2. **Goal:** Calculate

1. The problem is to find the value of $\cos 10^\circ$. 2. The cosine function is a trigonometric function that gives the ratio of the adjacent side to the hypotenuse in a right tr

1. Să se calculeze: a) sin 108°, cos 45°, cos 405°, tg 165°. 2. Dacă $x \in (0, \pi/3)$, $y \in (\pi/6, \pi/2)$ și $\sin x = \frac{4}{5}$, $\cos y = \frac{3}{5}$.

1. **State the problem:** A girl 1.2m tall is standing 25m away from a tower 18m high. We need to find the angle of elevation from her eyes to the top of the tower.

1. **State the problem:** Given side lengths $b=24$, $c=31$, and angle $\angle B=21^\circ$, use the Law of Sines to find all possible triangles (if any) with angles $\angle A_1$, $

1. **State the problem:** Given a triangle with side $a=37$, side $c=40$, and angle $\angle A=35^\circ$, find the possible values of angles $\angle B_1$, $\angle B_2$, $\angle C_1$