📏 trigonometry

1. **Problem Statement:** Graph the polar curve given by the equation $$r = 4 + 4 \cos \theta$$. 2. **Identify the type of curve:** The general form of a limaçon is $$r = a + b \co

1. The problem states: Prove that $\cos \frac{\pi}{7} \cos \frac{2\pi}{7} \cos \frac{4\pi}{7} = -\frac{1}{8}$. 2. We use the product-to-sum and symmetry properties of cosine to eva

1. **Problem statement:** (i) Calculate the length of |AB| in the triangular truss where |AC|=6 m and the roof pitch angle at A is 35°.

1. **State the problem:** We are given the function $$f(x) = -3 \sin\left(\frac{1}{2}(x + \pi)\right) - 1$$ and need to find its amplitude, period, vertical shift (V.S.), and phase

1. Berechne sin 50° + sin 140° + sin 230° + sin 320°: $$\sin 50^\circ \approx 0{,}77$$

1. Berechne sin 50° + sin 140° + sin 230° + sin 320°: $$\sin 50^\circ \approx 0{,}7660$$

1. **Problem statement:** Calculate the following trigonometric problems:

1. Bestimme den zweiten Winkel \(\alpha\) für die Gleichungen: **4c)** \(\sin \alpha = \sin 5^\circ\)

1. Problem statement: Find the second angle $\alpha$ for the given trigonometric equations as in examples c) and d). 2. Important formulas and rules:

1. **State the problem:** Find the amplitude, period, phase shift, and vertical shift of the function $$y = 3 + 4 \sin\left(x - \frac{\pi}{2}\right)$$ and sketch the graph for $$0

1. **Problem statement:** Find the exact expressions for the three primary trigonometric ratios (sine, cosine, and tangent) for angle $A$ in standard position. 2. **Recall definiti

1. **State the problem:** We need to find the length of the cable (side $AB$) in triangle $ABC$ where $AC=105$ m, angle $\angle BAC=12^\circ$, and the hill's inclination angle $\an

1. **State the problem:** Find the amplitude, period, phase shift, and vertical shift of the function $$y = 6 \sin x + 2$$ and sketch the graph for $$0^\circ \leq x \leq 360^\circ$

1. **State the problem:** Find the amplitude, period, phase shift, and vertical shift for the function $$y = 6 \sin x + 2$$ and sketch the graph for $$0^\circ \leq x \leq 360^\circ

1. **State the problem:** We are given that $m\angle A = 90^\circ$, $\cos B = 6x = 0.42$, and $\sin C = 10x = 0.34$. We need to solve for $x$. 2. **Analyze the given equations:**

1. **State the problem:** We have a triangle with one side length 5, and two angles 46° and 29°. We want to find the length of side $x$ using the Law of Sines. 2. **Recall the Law

1. **State the problem:** Solve the equation $$\sin^2 x - 5 \sin x + 6 = 2$$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation equal to zero:

1. Le problème semble concerner la simplification ou la manipulation d'une expression impliquant le sinus, probablement $-\sin(x) + \sin(x)$. 2. La formule importante ici est la pr

1. **State the problem:** We need to find the angle $x$ in a right triangle $UTS$ where the right angle is at $T$. 2. **Given:** Side $TS = 4.3$, hypotenuse $US = 9.8$, and angle $

1. **State the problem:** Given that $\cos \theta < 0$ and $\sin \theta = -\frac{2}{3}$, find $\tan \theta$ and the coordinates of point $P = (x, y)$ on the unit circle. 2. **Recal

1. **Problem statement:** Express $\sin 325^\circ$ in terms of $p$ given that $\sin 35^\circ = p$. 2. **Recall the sine function property:**