📏 trigonometry

1. **State the problem:** We need to solve for $x$ in the equation $\sin 35^\circ = \frac{5}{x}$ and round the answer to the nearest tenth. 2. **Recall the sine definition:** In a

1. **State the problem:** We are given the function $$y = 1 + \frac{1}{2} \cos(4\Theta + \frac{\pi}{3})$$ and asked to analyze its values at specific points and understand its grap

1. **State the problem:** We want to prove or verify the identity:

1. **Problem statement:** Diane has a triangular backyard with two fence posts. The distances from the deck to the fence posts are 90 m and 82 m, and the angle between these two li

1. Das Problem lautet: Finde Paare von Kärtchen, die denselben Wert haben. 2. Wir verwenden die trigonometrischen Identitäten und Werte für spezielle Winkel, um die Werte zu vergle

1. **Problem statement:** A boy starts at point A and walks 0.4 km to point D, then turns right 90° and walks 0.3 km to point C. The angle between AB and AD is $\theta$, and the pe

1. **State the problem:** We are given a triangle with angle $C = 63^\circ$, side opposite $B$ is $5.5$, side opposite $A$ is $4.7$, and we want to find the measure of angle $B$ us

1. **State the problem:** We are given a triangle with angles $35^\circ$, $105^\circ$, and the side opposite the $105^\circ$ angle is 7 units long. We need to find the length of si

1. **State the problem:** Isaac's eye is 1.63 meters above the ground. He measures the angle of elevation to the top of a skyscraper as 17° while standing 294 meters away horizonta

1. **State the problem:** A boat is observing a lighthouse beacon 139 feet above water. The angle of elevation from the boat to the beacon is 11°. We need to find the horizontal di

1. The problem is to prove the trigonometric identity $$\sin^2(x) + \cos^2(x) = 1$$. 2. This identity is fundamental in trigonometry and comes from the Pythagorean theorem applied

1. **Problem:** Find the missing side $x$ in a right triangle with a $72^\circ$ angle, adjacent side 6, and hypotenuse $x$. 2. **Formula:** Use the cosine function since cosine rel

1. a) Der Verlauf der Sinuskurve wird durch die Werte von $y = \sin x$ für Vielfache von $\frac{\pi}{8}$ bestimmt. Die markanten Punkte sind: $$\sin\left(\frac{3\pi}{8}\right) = \s

1. **State the problem:** Prove that $$\cos^4 x - \sin^4 x = \cos 2x$$. 2. **Recall the formula:** The difference of squares formula states $$a^2 - b^2 = (a-b)(a+b)$$.

1. **State the problem:** Prove or simplify the expression $$\frac{\cos(u - v)}{\cos u \sin v} = \tan u + \cot v$$. 2. **Recall the cosine difference formula:**

1. **State the problem:** We need to verify or simplify the equation $$\frac{1 + \tan y}{1 + \cot y} = \frac{\sec y}{\csc y}$$.

1. The problem is to identify the function of the form $$y = A \cos(B(x - C)) + D$$ given the graph characteristics. 2. The general cosine function formula is $$y = A \cos(B(x - C)

1. **Problem:** Calculate $\sin \alpha$ given $\cos \alpha = \frac{12}{37}$ and $270^\circ \leq \alpha \leq 360^\circ$. 2. **Formula and rules:**

1. **Problem Statement:** Find the amplitude, period, phase shift, vertical translation, and range of the function $y = -4 \sin(2x - \pi)$, then graph it over at least one period.

1. **Problem Statement:** A surveyor is standing 115 feet from the base of the Washington Monument. The angle of elevation to the top of the monument is 78.3°. We need to find the

1. **State the problem:** Marcus is 14 feet away from a 36-foot flagpole and looks up at the top of the flagpole. We need to find the angle of elevation from Marcus to the top of t