📏 trigonometry

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

1. **Problem:** Find the value of $\sin 210^\circ$ without using a calculator. 2. **Formula and rules:** The sine function for angles greater than $180^\circ$ can be found using th

1. Stating the problem: Solve the trigonometric equations $$\sqrt{\frac{1 + \sin \Delta}{1 - \sin \Delta}} = \tan \left(\frac{\pi}{4} + \frac{\Delta}{2}\right)$$

1. **Problem statement:** A man stands 50 m from the foot of a pole. His eye level is 1.5 m above the ground. The angle of elevation to the top of the pole is 30º.

1. We are asked to find the zeros (nulpunt) of the function $y = \sin\left(x - \frac{\pi}{4}\right)$. 2. The zeros of the sine function occur where the argument is an integer multi

1. **Problem Statement:** Given that $\pi < \theta < \frac{3\pi}{2}$ and $\tan(\theta) = 4$, find $\cos(\theta)$ and $\sin(\theta)$. 2. **Recall the definition of tangent:**

1. The problem asks to create pictures for all angles, which typically means illustrating angles in a circle or coordinate system. 2. To represent all angles, we use the unit circl

1. **State the problem:** We need to find the angle $\theta$ such that $0^\circ \leq \theta \leq 90^\circ$ and satisfies the equation $$3 \sin\left(\frac{\theta}{2} - 20^\circ\righ

1. **State the problem:** Solve for $\theta$ in the equation $$3 \sin\left(\frac{\theta}{2} - 20^\circ\right) = 0.85$$ where $0^\circ \leq \theta \leq 90^\circ$. 2. **Isolate the s

1. Diberikan persamaan trigonometri: $$\sin^2 x - \cos^2 x + \sin x = 0$$ dengan syarat $$0^\circ < x < 360^\circ$$. 2. Kita gunakan identitas trigonometri: $$\sin^2 x + \cos^2 x =

1. **State the problem:** Solve the trigonometric identity or equation:

1. The problem is to analyze the function $h(t) = 9.25 \sin[0.2(t - 2.5)] + 12.25$. 2. This is a sinusoidal function of the form $h(t) = A \sin(B(t - C)) + D$, where:

1. **Problem statement:** We need to find a sine function that models the height $h$ of a Ferris wheel car above the ground as a function of time $t$. 2. **Given data:**