📏 trigonometry

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^

1. The problem is to understand the function $\cos(x)$ and its properties. 2. The cosine function is a fundamental trigonometric function defined as the ratio of the adjacent side

1. **Problem Statement:** Given a sine wave oscillating between -2 and 2 with x-axis labeled in multiples of $\frac{\pi}{3}$, and two full cycles from 0 to $\frac{10\pi}{3}$, find

1. The problem is to express a trigonometric function in the form $$y = a \cos\left(b\left(x - \frac{c}{b}\right)\right) + d$$. 2. This form is a transformation of the cosine funct

1. **Stating the problem:** We want to find the equation of a sinusoidal function that fits the given graph data points and characteristics. 2. **Observations from the graph:**

1. **Stating the problem:** We are given the function $y = 2 \cos x$ and want to understand its behavior and graph. 2. **Formula and explanation:** The cosine function $\cos x$ osc

1. **Problem 1:** Given triangle ABC with \(m\angle C = 60^\circ\), side \(BC = 5.5\) m, and \(m\angle A = 80^\circ\), find \(b\), \(c\), and the missing angle measure \(m\angle B\

1. **Problem Statement:** An airplane A is flying at an elevation of 5625 ft above a straight highway. Two cars P and Q are on opposite sides of the plane on the highway. The angle

1. **Problem statement:** (a) Find $\tan\left(\arcsin\left(\frac{8}{x}\right)\right)$.