📏 trigonometry

1. The problem is to solve the inequality $\cos x > -\frac{1}{2}$. 2. Recall that the cosine function has a range of $[-1,1]$ and is periodic with period $2\pi$. The inequality ask

1. The problem is to find the values of $x$ for which $\cos x > -\frac{\sqrt{2}}{2}$.\n\n2. Recall that $\cos x$ ranges between $-1$ and $1$. The value $-\frac{\sqrt{2}}{2}$ is app

1. The problem is to solve the inequality $\sin x < \frac{\sqrt{2}}{2}$.\n\n2. Recall that $\sin x$ is the sine function, which oscillates between $-1$ and $1$. The value $\frac{\s

1. The problem asks for the expression of $\tan 4\theta$ in terms of $\tan \theta$ or $\tan 2\theta$. 2. We use the double angle formula for tangent: $$\tan 2\alpha = \frac{2 \tan

1. Problem: Find the value of $\sqrt{4\sin^2 \left(\frac{\pi}{24}\right)}$. 2. Formula and rules: Recall that $\sqrt{a^2} = |a|$, so

1. The problem states: Given $\cos \theta = \frac{1}{2} \left(a + \frac{1}{a}\right)$, find the value of $\cos 5\theta$. 2. We use the multiple-angle formula for cosine: $$\cos 5\t

1. The problem is to simplify or work with the expression involving $\tan^2 A$, not $\tan 2A$. 2. Recall the identity for $\tan^2 A$: it is simply the square of $\tan A$, i.e., $\t

1. **Problem Statement:** Prove that $$\tan 2A \cdot \sec^2 (90^\circ - A) - \sin^2 A \cdot \csc^2 (90^\circ - A) = 1$$ 2. **Recall the formulas and identities:**

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: $$\frac{1 + \sin \theta}{\cos \theta} + \frac{\cos \theta}{1 + \sin \theta} = 2 \sec \theta$$