📏 trigonometry

1. **Problem statement:** We have a ladder leaning against a wall forming a right triangle. The foot of the ladder is 2 m from the wall, and the angle between the ladder and the gr

1. **Problem Statement:** Given a right triangle with vertices A, B, and C, where angle C is the right angle, find $\sin \angle B$ and $\cos \angle B$.

1. **State the problem:** Given the equation $$\frac{\sin \theta + \cos \theta}{\sin \theta - \cos \theta} = 7,$$ find the value of $$\tan \theta.$$\n\n2. **Recall the formula and

1. **State the problem:** Given $\csc \theta = \frac{5}{4}$ and $0^\circ < \theta < 90^\circ$, find the value of $\sec \theta$. 2. **Recall definitions:**

1. **Problem Statement:** We are given the function $y = \sec \theta \tan \theta$ and a graph showing this function between $\theta = -\frac{\pi}{4}$ and $\theta = \frac{\pi}{4}$.

1. Problem: A cone is 8 cm high and its vertical angle is 62°. Find the diameter of its base. 2. Formula: The vertical angle of the cone is the angle at the apex of the isosceles t

1. **Problem (i): Solve for $0 \leq \theta < 360^\circ$ the equation $9 \sin(\theta + 60^\circ) = 4$.** 2. **Step 1: Isolate the sine function.**

1. **State the problem:** We have a right triangle with angles 64° and 32°, and the base length is 6.4 meters. We need to find the height $u$ and the hypotenuse $v$. 2. **Recall th

1. **State the problem:** Calculate $\tan 60^\circ + \sin 30^\circ$. 2. **Recall the values:**

1. The problem is to understand the expression involving $\tan x$ and other symbols, but the input appears to be nonsensical or corrupted. 2. Since $\tan x$ is the tangent function

1. The problem is to verify the equation $$2\tan^{-1}\frac{1}{2} - \tan^{-1}\frac{1}{7} = \frac{\pi}{4}$$. 2. Recall the formula for the tangent of a difference: $$\tan(a - b) = \f

1. **State the problem:** Simplify the expression $$\frac{1}{\cot x + 1} + \frac{1}{\tan x + 1}$$. 2. **Recall the definitions:**

1. **Problem statement:** Prove the trigonometric identity:

1. **State the problem:** Prove or verify the identity $$\sec^6\theta + \tan^6\theta = 1 + 3 \tan^2\theta \sec^2\theta$$. 2. **Recall the fundamental identity:** We know that $$\se

1. **State the problem:** Verify the trigonometric identity $$\sec^6 \theta + \tan^6 \theta = 1 + 3 \tan^2 \theta \sec^2 \theta$$. 2. **Recall key identities:**

1. **State the problem:** We are given two sine wave functions:

1. Énoncé du problème : Trouver la valeur de $x$ telle que $\sin x = \frac{\sqrt{3}}{2}$. 2. Formule utilisée : La fonction sinus est périodique et pour $\sin x = \frac{\sqrt{3}}{2

1. **State the problem:** We have two masts of heights 20 m and 12 m. The line joining their tops makes an angle of 35° with the horizontal. We need to find the distance between th

1. The problem asks to prove that $\sin(\alpha + \beta) = \sin \gamma$. 2. Normally, the sine addition formula states that $\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \al

1. **Problem Statement:** Find the height of the building given a right triangle formed by the building, a car, and an observer. The angles from the base are 60° and 30°, and the h

1. **State the problem:** We need to find the number of solutions (members) to the equation $$4 \sin 5\theta = 3$$ where $$\theta \in [0, 3\pi]$$. 2. **Rewrite the equation:** Divi