📏 trigonometry

1. **Stating the problem:** Create and solve three trigonometric function problems. 2. **Problem 1:** Find $\sin(30^\circ)$.

1. The problem is to understand how to graph tangent functions. 2. The tangent function is defined as $y=\tan(x)$, which is the ratio of sine to cosine: $\tan(x) = \frac{\sin(x)}{\

1. The problem is to find the value of $\sin 50^\circ$. 2. The sine function gives the ratio of the length of the side opposite the angle to the hypotenuse in a right triangle.

1. The problem is to find the value of $\sin 30^\circ$. 2. The sine function relates an angle in a right triangle to the ratio of the length of the opposite side to the hypotenuse.

1. The problem is to find the value of $\sin 10^\circ$. 2. The sine function gives the ratio of the length of the side opposite the angle to the hypotenuse in a right triangle.

1. **State the problem:** We have two right-angled triangles ABD and BCD sharing altitude BD perpendicular to AC. Given AD = 5 m, DC = 14 m, and angle BAD = 53°, we need to find th

1. The problem asks to find an expression for $f(x)$ in the form $f(x) = a \cdot \sin(x - b)$ where $a$ is the amplitude and $b \in \mathbb{Z}$. 2. From the graph description, the

1. **State the problem:** An aircraft flies 500 km on a bearing of 100 degrees, then 600 km on a bearing of 160 degrees. We need to find the distance and bearing from the starting

1. **Problem statement:** A ship travels on a N 500 E course (meaning 50 degrees east of north). It travels until it is due north of a port that is 10 km due east of the starting p

1. **Problem Statement:** Solve the equation $\cos\theta = \frac{\theta}{2}$ for $\theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ graphically and find the approximate soluti

1. **State the problem:** A ship sails 200 km on a bearing of 243.7 degrees. We need to find how far south and how far west the ship has traveled. 2. **Understanding bearings:** Be

1. The problem involves finding the height of a tree using trigonometry. 2. Given: The distance from the instrument to the tree base is $9\sqrt{3}$ meters, and the angle of elevati

1. **Problem statement:** A ship sails 200 km on a bearing of 243.7 degrees. We need to find how far south and how far west the ship has traveled. 2. **Understanding bearings:** A

1. **Problem statement:** A ship sails 200 km on a bearing of 243.7 degrees. We need to find how far south and how far west the ship has traveled. 2. **Understanding the bearing:**

1. **Problem statement:** An airplane flies on a course of S 300 E for 150 km. We need to find how far south the plane is from its starting point. 2. **Understanding the direction:

1. The problem is to simplify or understand the expression $\sin^2 x$. 2. The notation $\sin^2 x$ means $(\sin x)^2$, which is the square of the sine of $x$.

1. **State the problem:** We need to prove the trigonometric identity $$\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta$$. 2. **Recall the sine difference fo

1. **Stating the problem:** Find the value of $\sin(\tan^{-1}(x))$ or understand the relationship between sine and tangent functions. 2. **Formula and rules:** Recall that $\tan(\t

1. **Stating the problem:** Prove or verify the identity:

1. The problem is to find the angle in degrees. 2. To find an angle in degrees, you typically use trigonometric functions or convert from radians.

1. مسئله: بررسی تساوی الف) \( A = \sin^{-1} x + \sin^{-1} y = \sin^{-1} \left( x \sqrt{1-y^2} + y \sqrt{1-x^2} \right) \) با شرط \( |\sin^{-1} x + \sin^{-1} y| \leq \frac{\pi}{2} \