📏 trigonometry

1. Énonçons le problème : Montrer que $$\cos a \cos b = -\frac{1}{2} [\cos(a + b) + \cos(a - b)]$$. 2. Rappelons la formule trigonométrique connue pour le produit de cosinus :

1. **Énoncé du problème :** Montrer que $$\cos(2a) = \frac{1 - \tan^2 a}{1 + \tan^2 a}$$. 2. **Formule de départ :** On sait que $$\cos(2a) = \cos^2 a - \sin^2 a$$.

1. **State the problem:** We need to find the cosine of angle $C$ in a right triangle where the hypotenuse is 45 and the side adjacent to angle $C$ is 28. 2. **Recall the cosine de

1. **State the problem:** We need to find the sine of angle $H$ in a right triangle $IHG$ where $\angle G$ is the right angle. 2. **Recall the sine definition:** In a right triangl

1. **State the problem:** We need to find the sine of angle $B$ in a right triangle $BCD$ where $\angle C = 90^\circ$. 2. **Recall the sine definition:** In a right triangle, $\sin

1. **State the problem:** We need to find all values of $x$ in the interval $0^\circ \leq x \leq 360^\circ$ such that $\sin x = -0.8176$. 2. **Recall the sine function properties:*

1. **State the problem:** We need to solve for side $b$ in a triangle using the Law of Sines. 2. **Given:** Angle $A = 102^\circ$, side $a = 27$ opposite angle $A$, angle $B = 28^\

1. **Problem Statement:** Find the exact value of $\cot S$ in simplest form for the right triangle $QRS$ with right angle at $R$. Given sides: $QR=5$, $RS=\sqrt{17}$, and hypotenus

1. **State the problem:** We need to find the value of $\sin M$ in a right triangle $\triangle M O N$ where the right angle is at vertex $N$. The sides given are $ON = 4$ and $MO =

1. **Problem statement:** We are given a right triangle EDF with a right angle at E, side ED = 7, and angle F = 49°. We need to find the length of side DF. 2. **Identify the sides:

1. **State the problem:** A ladder leans against a wall forming a 60-degree angle with the ground. The height it reaches on the wall is 30 meters. We need to find the length of the

1. **State the problem:** We have a right triangle HGF with a right angle at G. The side HG measures 23 units, angle F is 71°, and we need to find side HF labeled as $x$.

1. The problem asks to convert an angle from degrees to radians and express the answer as an exact fraction in terms of $\pi$. 2. The formula to convert degrees to radians is:

1. **State the problem:** We have a triangle with angles 30°, 105°, and an unknown angle, and sides opposite 105° labeled $\sqrt{2}$, opposite 30° labeled $a$, and the base side la

1. **Problem:** Find the exact value of $\sin\left(\frac{7\pi}{6}\right)$. 2. **Recall:** The reference angle for $\frac{7\pi}{6}$ is $\frac{7\pi}{6} - \pi = \frac{\pi}{6}$.

1. The problem asks for the value of $\sec\left(\frac{\sqrt{3}}{2}\right)$.\n\n2. Recall that $\sec(x) = \frac{1}{\cos(x)}$. So, we need to find $\cos\left(\frac{\sqrt{3}}{2}\right

1. The problem asks to determine the secant of $\frac{\sqrt{3}}{2}$.\n\n2. The secant function is defined as the reciprocal of the cosine function: $$\sec(\theta) = \frac{1}{\cos(\

1. **State the problem:** We need to find the angle $x$ in a right triangle where the side adjacent to $x$ is 8.4 and the hypotenuse is 5. 2. **Identify the trigonometric function:

1. **State the problem:** Simplify the expression $$\frac{4 \sin 30^\circ - \tan 45^\circ}{2 \cos 30^\circ}$$ and express it in the form $$\tan x$$ where $$x$$ is an acute angle. 2

1. The problem is to simplify the expression $$\frac{(\sin x)^2}{\cos x}$$. 2. Recall the Pythagorean identity: $$\sin^2 x + \cos^2 x = 1$$.

1. The problem is to understand and express \(\tan x\) squared, which is commonly written as \(\tan^2 x\). 2. The notation \(\tan^2 x\) means \((\tan x)^2\), which is the square of