📏 trigonometry

1. **State the problem:** We need to find the length $m$ of the side opposite the $39^\circ$ angle in a right-angled triangle where the hypotenuse is 60. 2. **Formula used:** In a

1. **State the problem:** We need to find the function $f(x) = 2 \sin(kx) + d$ that matches the given sinusoidal wave with amplitude 2, passing through the point $(0,-1)$, and x-ax

1. **State the problem:** Find the exact value of $\sin\left(-\frac{11\pi}{4}\right)$.\n\n2. **Recall the sine function properties:** The sine function is periodic with period $2\p

1. **State the problem:** Prove the identity $$\frac{\cos\left(\frac{\pi}{2} + x\right)}{\cos(\pi + x)} = \tan x.$$\n\n2. **Recall the trigonometric angle addition formulas:**\n- $

1. **Problem:** Solve $\sin x = -\frac{\sqrt{2}}{2}$ on $[0, 2\pi)$. 2. **Formula and rules:** The sine function equals $-\frac{\sqrt{2}}{2}$ at angles where the reference angle is

1. The problem involves understanding the angle $\theta$ in a right triangle ramp, where $\theta$ is the angle between the horizontal base and the inclined ramp. 2. In a right tria

1. **State the problem:** Prove the identity $$\frac{\cos^2 x - \sin^2 x}{\cos x - \sin x} = \cos x + \sin x.$$\n\n2. **Recall the formula:** The numerator is a difference of squar

1. **State the problem:** Simplify and verify the expression $$\frac{\cos^2 x - \sin^2 x}{\cos x - \sin x} = \cos x + \sin x.$$\n\n2. **Rewrite the numerator using a trigonometric

1. **Aufgabe 10:** Zwei Winkel haben denselben Kosinuswert, wenn sie zueinander komplementär bezüglich 360° sind, also $\cos(\theta) = \cos(360^\circ - \theta)$. Gegebene Winkel: 2

1. **Problem 10:** Zwei Winkel mit demselben Kosinuswert finden. Gegeben sind Winkel: 220°, 200°, 250°, 10°, 170°, 140°, 110°, 350°, 60°, 190°, 210°, 150°, 300°, 160°.

1. **Énoncé du problème :** Déterminer la valeur de $x$ dans chaque triangle rectangle donné, en arrondissant au dixième près.

1. The problem asks to find the value of the expression $\cos^{-1} 0$. 2. The function $\cos^{-1} x$ (also called arccosine) gives the angle $\theta$ in radians whose cosine is $x$

1. **Problem statement:** We have a right-angled triangle ABC with |AB| = $h$ m, |BC| = 6 m, and point D on BC such that |BD| = 1 m and |DC| = 5 m. Given $\angle CAD = 45^\circ$ an

1. **State the problem:** We need to find the period of the sinusoidal function based on the graph description. 2. **Recall the period definition:** The period of a sinusoidal func

1. The problem asks to find the amplitude of the sinusoidal function shown in the graph. 2. The amplitude of a sinusoidal function is the distance from the midline (average value)

1. El problema pide calcular las razones trigonométricas directas (seno, coseno y tangente) de varios ángulos dados, relacionándolos con ángulos del primer cuadrante. 2. Regla impo

1. **State the problem:** Solve the equation $$\sin^3 x (1 + \cot^2 x) = 1$$ for $$0 \leq x < 2\pi$$ using the unit circle. 2. **Recall the Pythagorean identity:** $$1 + \cot^2 x =

1. **State the problem:** Simplify the expression $$\frac{\sec^2 x - 1}{\sin^2 x}$$ and identify it as a single trigonometric identity. 2. **Recall relevant identities:**

1. **State the problem:** We are given the function $$y = \sqrt{3} \sin x - 3 \cos x + 4$$ and want to understand its transformation from the basic sine function $$y = \sin x$$.

1. The problem is to graph the function $$y = 3 \sin \left( \frac{1}{2} x + \frac{\pi}{3} \right) - 1$$ which is a sine wave with transformations. 2. The general sine function is $

1. **Problemstellung:** Zeichne die Sinus- und Kosinuskurve mit verschiedenen Einheiten und berechne y-Werte für gegebene Winkelbereiche. 2. **Formeln und Regeln:**