📏 trigonometry

1. **State the problem:** Given $\tan \alpha = \frac{x}{x + 1}$ and $\tan \beta = \frac{1}{2x + 1}$, find the value of $\alpha + \beta$.

1. **Problem Statement:** Find the value of $$\cos \left( \frac{\pi}{6} - x \right) \cos \left( \frac{\pi}{3} - y \right) - \sin \left( \frac{\pi}{6} - x \right) \sin \left( \frac{

1. **State the problem:** We are given the function $$y=1-3\sin2\left(x+\frac{\pi}{3}\right)$$ and need to find its period, the interval for one period, the starting point of the i

1. **State the problem:** We are given the function $$y=1-3\sin2\left(x+\frac{\pi}{3}\right)$$ and we want to analyze it. 2. **Formula and explanation:** This is a sinusoidal funct

1. **State the problem:** We are given the function $$y=-2\sin\frac{\pi}{2}\left(x-\frac{1}{4}\right)$$ and need to find its period, interval, starting point, and 5 x-values (in ra

1. **Problem Statement:** We have a right triangle with a right angle at the top-left vertex. The side opposite angle $A$ (bottom-left) is $7.1$, and the hypotenuse (slanted side)

1. **State the problem:** We have a vertical door of height $88$ inches and an awning forming an angle of $50\degree$ with the wall. We want to find the length $L$ of the awning th

1. **State the problem:** We need to find the length $L$ of an awning that forms a $50^\circ$ angle with a vertical wall that is $88$ inches tall. 2. **Understand the triangle:** T

1. **State the problem:** We need to find all values of $\theta$ such that $$\tan \theta + \sqrt{3} = 0$$

1. **State the problem:** Identify which graph corresponds to the function $y = \sin(\theta)$. 2. **Recall the properties of $y = \sin(\theta)$:**

1. **State the problem:** Find the value of $$\cos\left(\sin^{-1}\left(-\frac{\sqrt{2}}{2}\right)\right)$$.

1. **Problem:** Calculate length $a$ in a right triangle with base $46$ m and vertical side $30$ m. 2. **Formula:** Use Pythagoras theorem: $$a = \sqrt{46^2 + 30^2}$$

1. **Problem statement:** We have a right triangle with sides opposite to angle $\theta$ as 3, adjacent to $\theta$ as 4, and hypotenuse as 5. We need to find $\cos(\theta)$.\n\n2.

1. **Problem 9: Evaluate $\sin^{-1}(\sin \frac{5\pi}{4})$.** 2. The angle $\frac{5\pi}{4}$ radians equals $225^\circ$, which lies in the third quadrant.

1. **Problem Statement:** (a) Express $$2 \cos \theta + 8 \sin \theta$$ in the form $$R \cos (\theta - \alpha)$$ where $$R > 0$$ and $$0 < \alpha < \frac{\pi}{2}$$. Find exact $$R$

1. **Problem Statement:** Given the first three terms of a geometric series: $$12 \cos \theta,$$ $$5 + 2 \sin \theta,$$ and $$6 \tan \theta,$$ show that $$4 \sin^2 \theta - 52 \sin

1. **State the problem:** We need to find the distance between two points, A and B, where a boat measures angles of elevation to a lighthouse beacon 121 feet above water as $13^\ci

1. **Problem Statement:** Meena is observing a boat from a lighthouse $120$ feet above water. The angles of depression to the boat at two points, A and B, are $18^\circ$ and $44^\c

1. **State the problem:** Jeremiah is $126$ feet above water in a lighthouse. He measures angles of depression to a boat at two points, A and B, as $11^\circ$ and $78^\circ$ respec

1. **Problem Statement:** Jeremiah is $126$ feet above water in a lighthouse. He measures angles of depression to a boat at two points, A and B, as $11^\circ$ and $78^\circ$ respec

1. **State the problem:** We want to prove the trigonometric identity $$\sin 3a = 3 \sin a - 4 \sin^3 a.$$\n\n2. **Recall the formula for sine of a sum:** $$\sin(x+y) = \sin x \cos