📏 trigonometry

1. **State the problem:** Find the exact value of $\cos(-510^\circ)$. 2. **Recall the cosine function property:** Cosine is an even function, so $\cos(-\theta) = \cos(\theta)$. The

1. The problem is to solve the equation involving the tangent function: $\tan\left(\frac{\pi}{2} x\right)$ and $2 \tan\left(t - \frac{\pi}{2}\right)$. We need to understand what is

1. The problem is to understand the general formula for $\sin(\pi y)$.\n\n2. The sine function $\sin(x)$ is periodic with period $2\pi$, meaning $\sin(x + 2\pi) = \sin(x)$.\n\n3. T

1. The problem asks to write the ratios equal to $\sin A$ and $\cos A$ for the given right triangle. 2. Recall the definitions for sine and cosine in a right triangle:

1. **State the problem:** Solve the equation $$\cot 2x = - \sin x \cdot \cos x$$. 2. **Recall the formulas:**

1. Stating the problem: Solve the equation $$9 \cos(2\theta) + 3 = 5 \cos \theta$$ for $$\theta$$. 2. Recall the double-angle identity for cosine: $$\cos(2\theta) = 2\cos^2\theta -

1. **Stating the problem:** We are given several trigonometric identities and functions involving angle $\omega$ and variable $x$, and we need to analyze and solve the expressions

1. **State the problem:** We have a right triangle formed by the central pole (20 ft), the slant height (26 ft), and the base of the tent. We want to find the angle between the cen

1. **State the problem:** Calculate the size of angle BAC in each right-angled triangle using the tangent ratio. 2. **Recall the formula:** For a right triangle, \( \tan \theta = \

1. **Problem statement:** Find the trigonometric values of $\sin \omega$ and $\cos \omega$ given the point $M(-0.8, 0.6)$ on the unit circle in the second quadrant, where $\omega =

1. **State the problem:** Simplify the expression $$\frac{1+\tan^2(x)}{1+\cot^2(x)}$$. 2. **Recall the Pythagorean identities:**

1. مسئله: حل معادله $\tan x = 1 + \tan^2 x$ برای قسمت ب. 2. فرمول و قوانین مهم: می‌دانیم که $\tan^2 x + 1 = \sec^2 x$، پس معادله را می‌توان به صورت $\tan x = \sec^2 x$ نوشت.

1. **State the problem:** Calculate the value of $$\sin\left(-\frac{\pi}{12}\right) \cdot \csc\left(\frac{25\pi}{12}\right)$$. 2. **Recall definitions and properties:**

1. **State the problem:** Simplify the expression $\tan 40^\circ - \frac{\sin 40^\circ}{\cos 40^\circ}$. 2. **Recall the formulas:**

1. **State the problem:** We are given the function $y = -2 \cos(2x + 2\pi)$ and need to find its amplitude, period, and phase shift. 2. **Amplitude:** The amplitude of a cosine fu

1. **State the problem:** We have a flagpole on top of a building that is 9.0 m tall. From the top of the flagpole, the angle of depression to a spot on the ground is 49°. 2. From

1. **State the problem:** A construction worker 6.5 feet tall stands 8 feet away from a two-storey building 40 feet high. We need to find the angle of elevation from the worker's e

1. **State the problem:** We have a right triangle HIJ with a right angle at vertex I. Angle H is 27°.

1. **State the problem:** We need to find the angle $x$ in a right triangle with a right angle at $P$. The side opposite angle $x$ (OP) is 56, and the side adjacent to angle $x$ (P

1. **State the problem:** Simplify the expression $2\sin^2 x + \sin x - \cos^2 x$. 2. **Recall the Pythagorean identity:** $\sin^2 x + \cos^2 x = 1$.

1. **State the problem:** We need to graph a sine function with amplitude 4, period $\pi$, midline $y=-3$, and y-intercept at $(0,-3)$. The graph is not reflected over the x-axis.