📏 trigonometry

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

1. **Problem statement:** A surveyor stands at point A. The true bearing of a tree from A is 070°.

1. **Problem:** An airplane flies on a bearing of 120° for 200 km, then turns to a bearing of 210° for 150 km. Calculate its distance from the starting point. 2. **Formula and rule

1. **Problem statement:** Two cars start from the same intersection where roads meet at 34°. The slower car travels at 80 km/h, the faster at 100 km/h. After 2 hours, a helicopter

1. The problem states: Given $\sin 2\theta = \frac{7}{11}$, find the value of $\sin 2\theta$. 2. The formula for the sine of a double angle is $\sin 2\theta = 2 \sin \theta \cos \t

1. **Stating the problem:** Verify if the equation $$\frac{\sin x}{1 + \cos x} + \frac{\cos x}{\sin x} = \csc x$$ is true.

1. **Problem statement:** A boat sails 8 km north from point P to Q, then 6 km west from Q to R. We need to find the bearing of R from P and the distance from P to R. 2. **Formula

1. **State the problem:** Solve the trigonometric equation $$2\cos^2(x) - 3\cos(x) + 1 = 0$$ for $$x$$ in the interval $$[0, 2\pi]$$. 2. **Identify the formula and substitution:**

1. **State the problem:** Solve the trigonometric equation $$2\cos^2 x - 3\cos x = 0$$ for $x$. 2. **Formula and rules:** This is a quadratic equation in terms of $\cos x$. We can

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

1. **Problem statement:** We have two tracking stations A and B on the ground, 50 miles apart horizontally. The angles of elevation to a satellite from A and B are 87.0° and 84.2°,

1. **Problem 1: Solve triangle ABC with given $A=38^\circ40'$ , $a=9.72$ m, and $b=11.8$ m.** 2. Use the Law of Sines: $$\frac{a}{\sin A} = \frac{b}{\sin B}$$

1. The problem is to provide the formulas for SOHCAHTOA, the sine law, and the cosine law. 2. SOHCAHTOA is a mnemonic to remember the definitions of sine, cosine, and tangent in a