📏 trigonometry

1. **Problem:** Find the sine, cosine, and tangent of a 60-degree angle. 2. **Formulas:**

1. Let's start with the basics of trigonometry. Trigonometry studies the relationships between the angles and sides of triangles, especially right triangles. 2. The primary functio

1. Let's start by understanding what trigonometry is. Trigonometry is the branch of mathematics that studies the relationships between the angles and sides of triangles, especially

1. Let's consider the problem: Find the exact value of $\sin(45^\circ)$ using trigonometric identities. 2. The formula we use is the sine of a sum identity: $$\sin(a+b) = \sin a \c

1. **Problem:** Solve the equation $$\sin^2 x - \cos x - 1 = 0$$. 2. **Formula and rules:** Use the Pythagorean identity $$\sin^2 x = 1 - \cos^2 x$$ to rewrite the equation in term

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

1. **Problem Statement:** Evaluate the expression $2 \sin(34^\circ) \cos(34^\circ)$.\n\n2. **Formula Used:** The double-angle identity for sine states that: $$2 \sin(\theta) \cos(\

1. The problem is to find the value of a trigonometric expression or solve a problem involving angles without using the sine function, suitable for a grade 9 level. 2. Since sine i

1. **Problem Statement:** Evaluate the given inverse trigonometric expressions and trigonometric values involving inverse functions. 2. **Recall important formulas and rules:**

1. **Problem statement:** Romeo and Paris are observing Juliet's balcony from two points 100 m apart. Romeo sees the balcony at an angle of elevation of 20° facing north, and Paris

1. **Problem:** Work out the length of the missing side of the triangles given. 2. **Formula:** Use the Pythagorean theorem for right triangles: $$A^2 + B^2 = C^2$$ where $C$ is th

1. **State the problem:** Simplify and verify the identity $$\frac{\sin \theta}{1 - \cos \theta} - \cot \theta = \csc \theta.$$ 2. **Recall formulas and identities:**

1. نبدأ بكتابة المعطى: \( \tan x = 4 \tan 45 \tan 45 \) حيث \( x \) زاوية حادة. 2. نعرف أن \( \tan 45 = 1 \) لأن ظل 45 درجة يساوي 1.

1. **State the problem:** Solve the equation $$\frac{4}{r} = \cos 43^\circ$$ for $r$ and give the answer to 2 decimal places. 2. **Formula and rules:** To isolate $r$, multiply bot

1. Problem: Find the coordinates of the point $E(t)$ on the unit circle and determine the sine and cosine values for $t = \frac{321\pi}{2}$ (part a) and $t = \frac{141\pi}{2}$ (par

1. Problem: Calculate the values for parts a) and f) of each given trigonometric task. 2. We will solve the first problem from each set (a) and the last problem (f) as requested.

1. **Problem:** Draw the angle $-230^\circ$ and label its initial and terminal sides. 2. **Understanding angles:** Angles are measured from the positive x-axis (initial side) count

1. Let's start with a basic trigonometry problem: Find $\sin(30^\circ)$. 2. The formula for sine in a right triangle is $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$.

1. **State the problem:** Simplify the expression \( \frac{\cos(\theta)}{1 - \sin(\theta)} - \tan(\theta) \). 2. **Recall formulas and identities:**

1. **Problem statement:** A surveyor in an airplane observes the angle of depression to the near side of a lake as 45° and to the far side as 32°. The airplane is 9750 m from the n

1. **State the problem:** Simplify and analyze the expression $80\sin(\theta) + 20$. 2. **Formula and rules:** The sine function $\sin(\theta)$ oscillates between $-1$ and $1$. Mul