📏 trigonometry

1. **Problem 53:** Given $\tan x = \frac{3}{4}$ and $0 < x < 90^\circ$, evaluate $\frac{\cos x}{2 \sin x}$. 2. **Recall the definitions and relationships:**

1. مسئله اول: مقدار عبارت $$A=\frac{\sin\left(\frac{\pi}{v}\right)+\sin\left(\frac{2\pi}{v}\right)+\sin\left(\frac{3\pi}{v}\right)}{\sin\left(\frac{4\pi}{v}\right)+\sin\left(\frac{

1. مسئله اول: اگر $6 = \tan x + \cot x$ باشد، مقدار عبارت $$A = \frac{\tan^7 x + \cot^7 x}{\sin^9 x + \cos^9 x}$$

1. The problem asks for the value of the angle $\theta$ when both $x$ and $y$ coordinates are negative. 2. In the Cartesian coordinate system, the angle $\theta$ is typically measu

1. **مسئله اول:** مقدار عبارت $$A = \frac{\sin \frac{7 \pi}{3} - \cos \frac{\Delta \pi}{6}}{\sin \left( \frac{-13 \pi}{3} \right) + \frac{1}{2} \tan \left( -\frac{4 \pi}{3} \right)

1. The problem is to create a table of values for the sine (sin), cosine (cos), and tangent (tan) functions. 2. These are trigonometric functions defined for an angle $\theta$ (in

1. The problem is to find the values of the trigonometric functions sine (sin), cosine (cos), and tangent (tan) for common angles. 2. The main angles we consider are $0^\circ$, $30

1. The problem asks if $\tan^{-1}(1)$ equals $\frac{\pi}{4}$.\n\n2. The function $\tan^{-1}(x)$, also called arctangent, gives the angle whose tangent is $x$.\n\n3. We know from tr

1. **Problem Statement:** We are given triangle ABC with angle $B = 120^\circ$, side $a = 12$ opposite vertex $C$, and side $c = 12$ opposite vertex $A$. We need to find the length

1. **Problem Statement:** We are given a triangle with vertices A, B, and C. The angle at vertex B is $120^\circ$. The sides opposite vertices A and C are both 12 units long, and w

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

1. **Prove the identity:** Given expression:

1. The problem asks for the sine of the value $$-\frac{1}{2}\sqrt{7}$$. 2. First, recognize that $$\sqrt{7}$$ is approximately 2.64575, so $$-\frac{1}{2}\sqrt{7} \approx -\frac{1}{

1. The problem is to find the value of $\sin(10)$ where 10 is in degrees. 2. Recall that the sine function takes an angle and returns the ratio of the length of the opposite side t

1. The problem is to find the sine of the number 10. 2. We assume the number 10 is in radians since no unit is specified.

1. **State the problem:** We are given a triangle ABC with angles $\angle A = 98.4^\circ$, $\angle B = 24.6^\circ$, and side $AC = 400$ units. We need to find side $x = AB$, which

1. **State the problem:** We are given a sinusoidal graph of $f(x)$ with amplitude 9 and need to find the amplitude, period, and a formula for $f(x)$ in the form $A\cos(Bx)$, $A\si

1. **State the problem:** We want to find the limit as $n \to \infty$ of the sequence $$S_n = \sin \left( \frac{(n+1) \pi}{12n+11} \right).$$

1. **State the problem:** We want to evaluate the limit $$\lim_{n \to \infty} S_n$$ where $$S_n = \sin\left(\frac{(n+1)\pi}{12n + 11}\right).$$ 2. **Analyze the argument of the sin

1. The problem is to find the value of $x$ such that $\cos x = 6$. 2. Recall that the cosine function, $\cos x$, has a range of values between $-1$ and $1$ for all real numbers $x$

1. **State the problem:** We have a right-angled triangle with one side (adjacent to the 37° angle) measuring 1.95 m and we want to find the hypotenuse length $r$. 2. **Identify th