📏 trigonometry

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:**

1. **Problem:** Find the exact value of $\cos\left(\frac{11\pi}{6}\right)$.\n\n2. **Formula and rules:** The cosine function on the unit circle corresponds to the $x$-coordinate of

1. **Problem Statement:** Given \(\cos 30^\circ = \frac{\sqrt{3}}{2}\) and \(\cot 30^\circ = \sqrt{3}\), find \(\sec 30^\circ\) and \(\tan 30^\circ\). 2. **Recall the definitions:*

1. **Problem Statement:** Find $\sec 30^\circ$ given that $\cos 30^\circ = \frac{\sqrt{3}}{2}$ and $\cot 30^\circ = \sqrt{3}$.\n\n2. **Recall the definition:** $\sec \theta = \frac

1. **State the problem:** Given $t = \frac{8\pi}{3}$, evaluate $\sin t$, $\cos t$, $\tan t$, $\csc t$, $\sec t$, and $\cot t$. 2. **Recall the periodicity and reference angle:** Th

1. The problem is to evaluate $\csc 270^\circ$ without using a calculator. 2. Recall that $\csc \theta = \frac{1}{\sin \theta}$, so we need to find $\sin 270^\circ$ first.

1. **State the problem:** Evaluate $\sec \frac{\pi}{4}$ without using a calculator. 2. **Recall the definition:** The secant function is the reciprocal of the cosine function, so

1. **Problem Statement:** Evaluate $\sin(-420^\circ)$ and also find $\cos(-420^\circ)$, $\tan(-420^\circ)$, $\csc(-420^\circ)$, $\sec(-420^\circ)$, and $\cot(-420^\circ)$ without a

1. **State the problem:** We have triangle ABC with sides AB = 9.7 m, BC = 12.3 m, and angle ABC = 115°. We need to find angle ACB, labeled as $x$, correct to 3 significant figures

1. **Problem statement:** Given that $\tan \theta$ is undefined and $8\pi \leq \theta \leq 9\pi$, find $\sin \theta$, $\cot \theta$, $\cos \theta$, and also find $\csc \theta$ and