📏 trigonometry

1. The problem asks us to solve the trigonometric expression: $$\frac{\sin 30^\circ \tan 45^\circ}{\sin 90^\circ} = ?$$

1. Problem (e): Find hypotenuse $x$ of a right triangle with angle $30^\circ$ and adjacent side 5 m. 2. Use cosine relation: $\cos(30^\circ)=\frac{\text{adjacent}}{\text{hypotenuse

1. **Problem (c):** Given an angle of 60° and the hypotenuse (or a side) of 20 meters, solve for the adjacent side or other relevant side. 2. Since the problem statement lacks expl

1. **State the problem:** Given the expression $$\tan C_1 = \frac{\tan d_2}{\tan d_1 \sin \Phi} - \cot \Phi,$$ where $$d_1 = 51^\circ 02' 00''$$, $$d_2 = 42^\circ 33' 00''$$, and $

1. **Stating the problem:** We have two right triangles each with an angle of 29°. - In the first triangle, the side adjacent to the 29° angle is 15, and the hypotenuse is $x$.

1. **Stating the problem:** We have a right triangle with one angle of 60° and the side opposite this angle is 8 units. We need to find the length of the side adjacent to this angl

1. **State the problem:** We have a right triangle with a hypotenuse of length 7, one angle measuring 35°, and we want to find the length of the side opposite to the 35° angle, den

1. The problem: Find the length $x$ in the right-angled triangle where the hypotenuse is 16 cm and one angle is 24° 2. In a right triangle, the side opposite an angle can be found

1. Problem: Find the length $x$ in each right-angled triangle given an angle and a side length. 2. Understand that in right-angled triangles, we can use trigonometric ratios (sine,

1. **State the problem:** We need to find the value of $\left(\frac{\csc 1^\circ}{\alpha}\right)^2$ where $$

1. **State the problem:** We have a right triangle with hypotenuse 105 ft and an angle of 30° between the ground and the hypotenuse. 2. The tower height is 85 ft, and the stack hei

1. Problem: Given $A=49.7^\circ$ and $B=67.2^\circ$, we need to find $\cos 3A + \sin B$, rounded to three decimal places. 2. Calculate $3A$:

1. Let's first write the expression clearly: $$\sum_{k=1}^3 \frac{\sin\left(\frac{k\pi}{12}\right) + \sin\left(\frac{(6-k)\pi}{12}\right)}{\cos\left(\frac{k\pi}{12}\right) + \cos\l

1. **State the problem:** Simplify the expression $$\frac{\cos(\theta)}{1 - \sin(\theta)} - \tan(\theta)$$. 2. **Rewrite the tangent in terms of sine and cosine:**

1. **State the problem:** Simplify the expression $$\frac{\cos(\theta)}{1 - \sin(\theta)} - \tan(\theta)$$. 2. Rewrite $\tan(\theta)$ as $\frac{\sin(\theta)}{\cos(\theta)}$ to have

1. **State the problem:** We need to find the value of $\sin\left(\frac{4\pi}{6}\right)$.\n\n2. **Simplify the angle:** Simplify the fraction inside the sine function.\n$$\frac{4\p

1. **Problema**: Calcular os valores das funções trigonométricas dadas: (a) $\sin\left(\frac{4\pi}{6}\right)$ (b) $\cos\left(\frac{8\pi}{6}\right)$ (c) $\tan\left(\frac{5\pi}{4}\ri

1. The problem is to understand the expression $\cos x \sin x$. 2. This is a product of the cosine and sine trigonometric functions.

1. **State the problem:** Simplify the trigonometric expression $\sin x \cos^2 x - \sin x$. 2. **Factor the expression:** Notice that $\sin x$ is common in both terms, so factor it

1. For the triangle with the hypotenuse labeled "hyp" (longest side), the side opposite the acute angle is labeled "opp" and the side adjacent to the acute angle is labeled "adj".

1. **Problem statement:** We want to prove that $$\cos^4\left(\frac{\pi}{8}\right) + \cos^4\left(\frac{3\pi}{8}\right) + \cos^4\left(\frac{5\pi}{8}\right) + \cos^4\left(\frac{7\pi}