📏 trigonometry

1. **State the problem:** Find the horizontal distance from the airplane to the landmark given the altitude and angle of depression.

1. **State the problem:** Alan is flying at an altitude of 2000 m. The angle of depression to a landmark on the ground is 78°. We need to find the horizontal distance from the airp

1. **State the problem:** Given $x=3\tan\theta$, we want to label all sides of a right triangle and find the sides in terms of $x$ and $\theta$. 2. **Recall the definition of tange

1. Das Problem lautet: Gegeben ist \(\cos(0{,}5)\) mit den Winkeln 60° und 300°, und der Rest ist falsch. Wir sollen den korrekten Wert und Zusammenhang bestimmen. 2. Die Kosinusfu

1. **State the problem:** Solve the inequality $$\frac{3\cos(x)}{\sqrt{2} - 2\sin(x)} \geq 0$$ for $x$. 2. **Understand the inequality:** A fraction is non-negative if its numerato

1. **State the problem:** We need to sketch the graph of the function $$y = \sin(4x) - 2$$ including two full periods and correct axis markings.

1. **State the problem:** We need to sketch the graph of the function $$y = \cos\left(\frac{x}{2}\right) + 1$$ including two full periods and correct axis markings.

1. **State the problem:** We need to find the length of segment $|DAB|$ using trigonometry. 2. **Identify the triangle and known elements:** Assume $DAB$ is a triangle or a segment

1. **State the problem:** Given that $\sin \alpha = \frac{3}{5}$ and $\frac{\pi}{2} < \alpha < \pi$, find the exact values of $\cos \frac{\alpha}{2}$ and $\tan \frac{\alpha}{2}$. 2

1. **State the problem:** We need to find the equation of a sinusoidal function with a given period of $\frac{8\pi}{3}$. 2. **Recall the formula for the period of sine or cosine:**

1. **State the problem:** We need to write an equation of the form $y = a \sin bx$ or $y = a \cos bx$ to describe the given sinusoidal graph. 2. **Identify key features from the gr

1. **State the problem:** We need to write an equation of the form $y = a \sin bx$ or $y = a \cos bx$ for a sinusoidal graph with amplitude 4, midline $y=0$, and period $\frac{4\pi

1. **State the problem:** We need to write the equation of a sine or cosine function that matches the given graph. 2. **Identify key features from the graph:**

1. **State the problem:** Solve the trigonometric equation $$\sin^2 x = 3 \cos^2 x$$ for $x$. 2. **Use the Pythagorean identity:** Recall that $$\sin^2 x + \cos^2 x = 1$$.

1. **Problem statement:** Given $\tan x = -\frac{2}{3}$ with $-\frac{\pi}{2} < x < 0$, find the exact values of: 8. $\cos 2x$

1. **Problem statement:** Find the exact value of \( \cos(u - v) \) given \( \csc u = -\frac{13}{12} \) with \( \pi < u < \frac{3\pi}{2} \), and \( \cos v = \frac{4}{5} \) where \(

1. **Problem:** Given the function $$f(x) = 3 \cos\left(\frac{\pi x}{6}\right) + 1$$ for $$0 \leq x \leq 12$$, answer the following: 2. **a. Amplitude:** The amplitude of a cosine

1. **Problem statement:** Derivate die Formel für den Sinus von 30 Grad. 2. **Wichtige Formel:** Sinus eines Winkels in einem rechtwinkligen Dreieck ist definiert als das Verhältni

1. **State the problem:** We need to find the equation of a sine function $f(x)$ with the following characteristics: - Centered around $y=4$ (vertical shift)

1. The problem is to find an equation for a sine graph with the following characteristics: midline $y = -2$, amplitude $1$, period $\frac{\pi}{2}$, and a maximum at $x = \frac{\pi}

1. **Problem statement:** Calculate angle \(\angle B\) in trapezium ABCD where \(AB=8\), \(DC=2.5\), \(AD=3.2\), and \(\angle A=60^\circ\).\n\n2. **Known facts and formula:** In tr