📏 trigonometry

1. **Problem Statement:** Prove the following trigonometric identities given angles $A=0^\circ$, $B=30^\circ$, $C=45^\circ$, $D=60^\circ$, and $E=90^\circ$:

1. **State the problem:** Solve the expression $$4A = \frac{8}{3} + \frac{2}{1} \cos 2A + \frac{8}{1} \cos 4A$$ for $A$. 2. **Rewrite the expression clearly:**

1. **Problem statement:** Prove that $$\cos^4 A = \frac{3}{8} + \frac{1}{2} \cos 2A + \frac{1}{8} \cos 4A.$$\n\n2. **Formula and identities used:** We use the double-angle identity

1. Stating the problem: We need to find the value of $\cos^2 40^\circ + 1$. 2. Recall the Pythagorean identity:

1. The problem is to graph the function $$y = \frac{\sin x}{x}$$ and understand its behavior. 2. This function is known as the sinc function (unnormalized). It is defined as $$y =

1. **State the problem:** We need to find the height $P$ of the lamp using the given measurements: the theodolite is 1.6 m tall, positioned 4 m from the lamp, and the angle of elev

1. **State the problem:** We need to find the height $H$ of the building given a right triangle where the lamp height is 3 m, the horizontal distance between the lamp and building

1. **Problem:** Given a triangle with angles 100° and 38°, and side opposite 100° is 13 cm, find side $x$ opposite 38°. 2. **Formula:** Use the Law of Sines: $$\frac{a}{\sin A} = \

1. **State the problem:** Simplify and solve the trigonometric equation $$\sin^2\theta \cot\theta \sec\theta = \sin\theta$$. 2. **Recall definitions and formulas:**

1. The problem asks for the x-coordinate of point A on a circle centered at the origin with radius 1, where point A is located at an angle $\frac{\pi}{5}$ radians above the negativ

1. The problem asks for the x-coordinate of point A, which lies on a circle centered at the origin with radius 1. 2. Point A is in the second quadrant, and the angle between the po

1. **Problem Statement:** Find the y-coordinate of point A on the unit circle at an angle of $\frac{3\pi}{2}$ radians. 2. **Relevant Formula:** On the unit circle, a point at angle

1. **State the problem:** Verify the trigonometric identity: $$\frac{1 + \sin \theta}{\cos \theta} + \frac{\cos \theta}{1 + \sin \theta} = 2 \sec \theta$$

1. **State the problem:** Verify the trigonometric identity $$\frac{\cos \theta}{1 + \sin \theta} + \frac{\sin \theta}{\cos \theta} = 2 \sec \theta$$

1. Problem statement: Solve $\sqrt[3]{2\cos^2 x} - \sqrt[3]{2\sin^2 x} = \sqrt[3]{\cos 2x}$ for $x\in[0,\pi]$. 2. Let $a=\sqrt[3]{2\cos^2 x}$, $b=\sqrt[3]{2\sin^2 x}$ and $c=\sqrt[

1. **State the problem:** We need to find the height $P$ of the lamp using the given measurements and the angle of elevation. 2. **Identify the known values:**

1. **State the problem:** We are given the function $f(x) = \frac{\cot x}{1 + \csc x}$ and need to simplify it. 2. **Recall the definitions:** \(\cot x = \frac{\cos x}{\sin x}\) an

1. The problem is to find the value of $\sec\left(\frac{5\pi}{6}\right)$.\n\n2. Recall that $\sec(\theta) = \frac{1}{\cos(\theta)}$. So we need to find $\cos\left(\frac{5\pi}{6}\ri

1. **Problem Statement:** We analyze the cotangent function $\cot \alpha$ defined on the interval $(0^\circ, 90^\circ)$ using a right-angled triangle and its properties. 2. **Defin

1. **Problem Statement:** We want to define the inverse cosine function, denoted as $\arccos(x)$, which requires restricting the domain of the cosine function to an interval where

1. **Stating the problem:** We need to find the radian values of the given arcsin expressions and match them to the correct radian measures from the second group.