📏 trigonometry

1. Find the exact value of $\sin \frac{2\pi}{3} + \cos \frac{7\pi}{6} + \tan \frac{5\pi}{3}$. - $\sin \frac{2\pi}{3} = \sin 120^\circ = \frac{\sqrt{3}}{2}$ (since sine is positive

1. The problem involves understanding the ratio of three fractions: $\frac{5}{7}$, $\frac{5\sqrt{3}}{7}$, and $\frac{10}{7}$, which are related to angles $45^\circ$, $-45^\circ$, $

1. **State the problem:** Verify the identity $\tan^2 x \sin^2 x = \tan^2 x - \sin^2 x$ by working only on one side. 2. **Choose the right side to simplify:**

1. **State the problem:** Simplify and verify the equation $\tan^2 x \sin^2 x = \tan^2 x - \sin^2 x$. 2. **Recall definitions and identities:**

1. **State the problem:** Find the angle $x$ such that $\cos x = \frac{1}{4}$.\n\n2. **Formula and rules:** The cosine function relates an angle to the ratio of the adjacent side o

1. **State the problem:** We need to find the angle $x$ such that $\cos x = 2$. 2. **Recall the range of cosine:** The cosine function outputs values only in the range $[-1, 1]$ fo

1. **State the problem:** We have a right triangle formed by the flat ground, the vertical height of the airplane, and the slant distance from point T to the airplane P. We need to

1. The problem asks for the trigonometric ratios \(\sin M\), \(\cos M\), and \(\tan M\) in the right triangle KLM with right angle at K. 2. Recall the definitions of the trigonomet

1. **State the problem:** We have a right triangle ABC with right angle at C.

1. **Problem 1:** Given triangle ABC with a right angle at B, angle at C is 40°, side AB = 22, find side AC = x. 2. Use the sine function in right triangle: \( \sin(\theta) = \frac

1. **State the problem:** We have a right triangle ABC with angle B as the right angle, angle A = 11°, side AB (adjacent to angle A) = 27, and hypotenuse AC = x. We need to find $x

1. **Problem 1:** Given a right triangle with an angle of 30°, hypotenuse 24 cm, and opposite side labeled 24 cm, solve for $x$ which is the adjacent side. 2. **Choosing the trigon

1. **State the problem:** We need to analyze and sketch the graph of the function $$y = -2 \cos\left(0.25x - \frac{\pi}{8}\right) + 3$$ by finding its amplitude, axis of the curve,

1. **State the problem:** We need to find the length of a ramp that is inclined at an angle of 15° from the ground. 2. **Identify the known values:** The angle of inclination $\the

1. **Problem statement:** We have a ski lift pole on a slope inclined at $21^\circ$ to the east. The shadow of the pole on the slope is 24 meters long. Sunlight comes from the west

1. The problem asks for the vertical shift of the function $$y = 4 \cos(x + \pi) - 2$$. 2. The general form of a cosine function with transformations is $$y = A \cos(B(x - C)) + D$

1. The problem asks for the phase shift of the function $$y = 4\cos\left(\frac{1}{2}x + \pi\right) - 2$$. 2. The general form of a cosine function with phase shift is $$y = A\cos(B

1. The problem asks for the phase shift of the function $$y = 4 \cos(x + \pi) - 2$$. 2. The general form of a cosine function with phase shift is $$y = A \cos(B(x - C)) + D$$, wher

1. **State the problem:** Find the period of the function $y = 4 \cos(5x)$.\n\n2. **Recall the formula for the period of cosine:** The general form is $y = A \cos(Bx)$, where the p

1. **State the problem:** We need to find two angles in radians between $-2\pi$ and $2\pi$ whose terminal sides pass through the origin and a given point on the circle in the recta

1. **State the problem:** We need to find the period of the functions $y = \sin(x)$ and $y = \cos(x)$. Both are trigonometric functions with well-known periodic behavior. 2. **Reca