📏 trigonometry

1. **Problem 1: Distance the second ship must sail** A ship travels west 50 km, then sails 52.4 km in direction E25°N. We want the direct distance from the harbour to the ship's fi

1. **Problem statement:** A tree is broken by the wind such that the broken part makes a 45° angle with the ground. The distance from the root to the point where the top touches th

1. **State the problem:** We have triangle ADE with sides AD = 60, AE = 146, and angle DAE = 55°. We need to find: (a) The bearing of A from E.

1. Δίνεται ότι $\tan \theta = \frac{4}{3}$ και $180^\circ < \theta < 270^\circ$. Θέλουμε να υπολογίσουμε την τιμή της παράστασης: $$A = \frac{4\cos \phi \theta - 5\sin \theta}{3\ta

1. **State the problem:** Find the smallest positive root of the equation $$\sin 3x = -1$$ where $x$ is in degrees. 2. **Recall the sine function properties:** The sine function eq

1. **Problem Statement:** You are on a sailing trip and want to find the angle of elevation to the top of a lighthouse from your boat. The triangle formed has two sides measuring 1

1. **State the problem:** We need to graph the function $f(x) = \csc x$ and discuss its behavior near $x=0$. 2. **Recall the definition:** The cosecant function is defined as $\csc

1. **Problem statement:** Sketch the function $f(x) = \cot(x)$ and mark its points of discontinuity. 2. **Formula and definition:** The cotangent function is defined as $\cot(x) =

1. **State the problem:** We are given the function $$s = \frac{4}{3\pi} \sin 3t + \frac{4}{5\pi} \cos 5t$$ and want to understand its behavior and graph. 2. **Formula and explanat

1. **Problem statement:** We have a right triangle where one leg (adjacent to the 45° angle) is 30 meters, and we want to find the height of the tower, which is the opposite leg. 2

1. **Problem statement:** Given $\cos \theta = \frac{12}{13}$, find $\sin \theta$. 2. **Formula used:** We use the Pythagorean identity:

1. **Problem Statement:** Given that $\tan \theta = \frac{7}{24}$, find $\sec \theta$. 2. **Recall the definitions and formulas:**

1. **State the problem:** We have a right triangle where the side adjacent to angle $\theta$ is 9 and the hypotenuse is 15. We need to find $\cos \theta$. 2. **Recall the formula:*

1. **State the problem:** We are given a right triangle with angle $\theta$, the side opposite $\theta$ is 6, and the hypotenuse is 10. We need to find $\sin \theta$. 2. **Recall t

1. **State the problem:** We are given two equations involving inverse trigonometric functions:

1. **State the problem:** Simplify the expression $$\frac{\cos(\theta)}{1 - \sin(\theta)} - \tan(\theta)$$. 2. **Recall formulas and identities:**

1. **State the problem:** Simplify the expression $$\frac{\cos(\theta)}{1-\sin(\theta)} - \tan(\theta)$$. 2. **Recall formulas and identities:**

1. **Problem statement:** A man at the top of a 75 m high tower sees two goats due west at angles of depression 10° and 17°. We need to find the distance between the two goats. 2.

1. **Problem statement:** Find the angle $x$ in each right-angled triangle given the sides. 2. **Formula and rules:** Use trigonometric ratios: sine, cosine, or tangent depending o

1. **State the problem:** Solve the equation $\sin x = -0.3$ for $x$. 2. **Recall the sine function properties:** The sine function has a range of $[-1,1]$ and is periodic with per

1. **Problem Statement:** Determine the values of scalars $A$ and $\omega$ for the trigonometric function $$y = A \sin(\omega x + 30^\circ)$$ given the graph of the wave for $$-90^