📏 trigonometry

1. **State the problem:** We are given a table of $x$ and $y$ values and need to find the equation of the periodic function that fits these points. 2. **Given table:**

1. The problem asks to write the equation of the periodic function based on the graph. 2. The graph shows a cosine wave with amplitude 4, oscillating between 4 and -4, and a period

1. **State the problem:** We need to find the equation of the periodic function based on the given graph. 2. **Identify the form:** The function is given as $f(t) = a \sin(bt)$ or

1. The problem asks to compare the amplitudes of the functions $f(x) = -1.8 \cos x$ and $g(x) = -3.6 \cos x$. 2. Recall that the amplitude of a cosine function $a \cos x$ is the ab

1. **State the problem:** We need to find what fraction of a semicircle the angle 315° represents and then convert 315° to radians. 2. **Recall the facts:**

1. **State the problem:** Given $\sin x = \frac{p - q}{p + q}$ where $x$ is between $0^\circ$ and $90^\circ$, find $1 - \tan^2 x$. 2. **Recall the identity:** We know that $1 - \ta

1. **Problem statement:** Prove that $\tan 20^\circ \times \tan 40^\circ \times \tan 80^\circ = \sqrt{3}$. 2. **Recall the tangent triple-angle identity:**

1. مسئله: اگر \( \sin \alpha = -\frac{1}{5} \) و \( \alpha \) زاویه‌ای در ربع سوم باشد، مقدار \( \sqrt{25 - \cot^2 \alpha} \) را بیابید. 2. فرمول‌ها و نکات مهم:

1. **State the problem:** Solve the trigonometric equation $$\cos^2 x + 2 \sin x + \sin^2 x = 0$$. 2. **Use the Pythagorean identity:** Recall that $$\cos^2 x + \sin^2 x = 1$$.

1. The problem is to find the value of $\cot(-300^\circ)$. 2. Recall that the cotangent function is the reciprocal of the tangent function:

1. **Problem statement:** Prove that $$\sin^{-1} \frac{3}{5} + \frac{1}{2} \cos^{-1} \frac{5}{13} - \cot^{-1} 2 = \tan^{-1} \frac{28}{29}$$. 2. **Recall formulas and identities:**

1. **State the problem:** Solve the equation $$\sin \theta + 2 \sin \theta \cos \theta = 0$$ for $$\theta$$ in the range $$0^\circ \leq \theta \leq 360^\circ$$. 2. **Write the equa

1. The problem is to prove the trigonometric identity: $$\sin 8A + \cos 8A = 1 - 2\sin^2 2A + \frac{1}{8} \sin^4 2A$$

1. The problem is to find the value of $\tan 75^\circ$. 2. We use the tangent addition formula: $$\tan(a+b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}$$

1. The problem is to find the value of $\tan 15^\circ$. 2. We use the angle subtraction formula for tangent:

1. **Problem statement:** Find the length of side $a$ in a right triangle with angles $35^\circ$, $55^\circ$, and $90^\circ$, where the side adjacent to the $55^\circ$ angle is 11

1. The problem asks to analyze and draw graphs of the given trigonometric functions: A: $y=6\sin(3t)$

1. **Problem Statement:** Identify the graph characteristics of the given sine functions: A: $y=8\sin\left(2t+\frac{\pi}{4}\right)$

1. مسئله: دوره تناوب تابع $$y=\cos\left(\frac{a}{b}x\right)$$ را بیابید، وقتی که تابع $$y=a\sin(bx)$$ دوره تناوبش مشخص است. 2. فرمول دوره تناوب برای تابع $$y=\sin(kx)$$ یا $$y=\cos

1. The problem asks us to find an expression for the function $f(x)$ in the form $f(x) = a \cdot \sin(x)$ or $f(x) = a \cdot \cos(x)$ where $a$ is an integer. 2. To determine $a$,

1. **Problem statement:** We need to find the height of the kite above the ground given the angle of elevation, the length of the string, and the sag in the string. 2. **Given:**