📏 trigonometry

1. مسئله 115: مقدار تابع $f(-\frac{1}{a})$ برای $f(x) = 2a \sin^r\left(\frac{\pi}{a} x\right)$ را پیدا کنید. تابع سینوسی در نقطه $x = -\frac{1}{a}$ داریم:

1. **Problem 1: Estimate the height of the building** A person views the top of a building from a point 30 m away, and the line of sight makes a 40° angle with the horizontal. We w

1. **State the problem:** Sadie is 1.69 meters tall and stands 275 meters from a skyscraper. She measures the angle of elevation to the top of the skyscraper as 36°. We need to fin

1. **Problem 1: Angle of Depression from Hot Air Balloon** We are given a hot air balloon 40 ft above the ground and 70 ft away horizontally from a farm. We need to find the angle

1. **Problem 1: Angle of Depression from Hot Air Balloon** A hot air balloon is 40 ft above the ground and 70 ft away horizontally from a farm. We need to find the angle of depress

1. **State the problem:** Given the equation $\sin 2A = 2 \sin A$, find the value(s) of $A$. 2. **Recall the double-angle formula for sine:**

1. **Problem 1: Height of the sand pile** The sand forms a cone with a base diameter of 8 m, so the radius $r$ is half of that: $r=\frac{8}{2}=4$ m.

1. **State the problem:** We need to write an equation for a cosine function with a period of $3\pi$ and a phase shift of $+\frac{\pi}{3}$. 2. **Recall the general form of a cosine

1. **State the problem:** Simplify the expression $\sin 52^\circ \sin 68^\circ - \sin 47^\circ \cos 77^\circ - \cos 65^\circ \cos 81^\circ$.\n\n2. **Recall trigonometric identities

1. **Problem statement:** Given $\cos \theta = -\frac{2}{5}$ and $\sin \theta > 0$, find the quadrant where $\theta$ lies, then find $\sin \theta$, $\tan \theta$, $\sec \theta$, $\

1. **Problem Statement:** Given $\tan \theta = -\frac{12}{5}$ and $\sin \theta > 0$, find the quadrant of $\theta$, then find $\sin \theta$, $\cos \theta$, $\sec \theta$, $\csc \th

1. **State the problem:** Given $\cot \phi = -6$ and $\sin \phi = \frac{\sqrt{37}}{37}$, find $\tan \phi$, $\csc \phi$, and $\sec \phi$. 2. **Recall the definitions and formulas:**

1. **State the problem:** Given $\tan(\theta) = -2\sqrt{2}$ and $\cos(\theta) > 0$, find the exact value of $\sin(\theta)$. 2. **Recall the identity:** We know that $\tan(\theta) =

1. **State the problem:** Given $\cos \theta = -\frac{2}{5}$ and $\sin \theta > 0$, find the quadrant of $\theta$ and the values of $\sin \theta$, $\tan \theta$, $\sec \theta$, $\c

1. **State the problem:** We need to find the quadrant(s) where the angle $\theta$ satisfies the inequalities: $$\cos(\theta) < 0$$

1. **Problem Statement:** Determine the quadrant in which angle $\theta$ lies given the inequalities: $$\sin(\theta) < 0$$

1. **Problem Statement:** Given the terminal side of angle $\theta$ lies on the line $-4x + 3y = 0$ in Quadrant IV, find the values of $\sin \theta$, $\cos \theta$, $\tan \theta$,

1. **Problem Statement:** Given the terminal side of angle $\theta$ lies on the line $-4x + 3y = 0$ in Quadrant IV, find $\sin \theta$, $\cos \theta$, $\tan \theta$, $\sec \theta$,

1. **State the problem:** Given $\cos \theta = \frac{6}{10} = 0.6$ and $\tan \theta < 0$, find the values of $\sin \theta$, $\tan \theta$, $\csc \theta$, $\sec \theta$, and $\cot \

1. **Problem statement:** Given point $P(-15, y)$ with $OP=17$ units and reflex angle $\angle MOP=\alpha$, find: 5.1.1 $y$

1. Problem 15: In triangle $\triangle ABC$, given $a=8$ cm, $b=5$ cm, and angle $C=60^\circ 2'$, find side $c$. 2. Use the Law of Cosines formula: $$c^2 = a^2 + b^2 - 2ab \cos C$$