📏 trigonometry

1. Stating the problem: We need to find an approximate value for $\frac{\cos 2^\circ}{\sin 1^\circ}$ given that $1^\circ=0.018$ radians and $(0.018)^2=0.000324$. The answer should

1. The problem is to understand the meaning of \(\tan \alpha\) based on the given graph which shows an angle \(\alpha\) between a line and the horizontal axis.\n\n2. By definition,

1. Let's clarify the difference between $\cos 2^\circ$ and $\cos^2 1^\circ$.\n 2. The expression $\cos 2^\circ$ means the cosine of $2$ degrees. It's simply the cosine function app

1. Problem: Find, in terms of $\theta$, an approximate value for $$\frac{\sin 4\theta + \tan 2\theta}{3 + \cos 2\theta}$$ for small values of $\theta$ (i.e., $\theta \to 0$) neglec

1. The problem asks to evaluate $\sin 2^\circ$, which is the sine of $2$ degrees. 2. Sine of small angles can be evaluated using a calculator or known sine values if available.

1. State the problem: Prove that $$\frac{\cot \theta + \tan \theta}{\sec \theta} = \csc \theta.$$\n\n2. Write the trigonometric functions in terms of sine and cosine:\n$$\cot \thet

1. **State the problem:** We want to prove the identity $$\frac{1-\sin\theta}{\cos\theta} = 3\cot\theta$$

1. The problem is to verify the trigonometric identity $\sin 2x = 2 \sin x \cos x$.\n\n2. Start with the double-angle formula for sine, which states that for any angle $x$,\n$$\sin

1. Stating the problem: We want to verify the trigonometric identity $$\tan(90^\circ + \theta) = -\cot \theta$$. 2. Recall the definition and properties:

1. The problem is to prove that $\sin^2 x + \cos^2 x = 1$ for any angle $x$. 2. Recall the Pythagorean identity from trigonometry which states that the square of the sine of an ang

1. Problem statement: Find the exact values of (a) $\tan(\frac{\pi}{3})$, (b) $\sin(\frac{7\pi}{6})$, and (c) $\sec(\frac{5\pi}{3})$. 2. For (a) $\tan(\frac{\pi}{3})$:

1. The problem is to solve the equation $2\cot(2a) = 3$ for $a$. 2. Start by isolating $\cot(2a)$:

1. The problem is to solve the equation $2\cot 2x = 3$ for $x$. 2. Start by isolating the cotangent term: $$\cot 2x = \frac{3}{2}$$

1. Problem 5: A ship sails 20 km due north and then 35 km due east. Find how far it is from its starting point. Step 1: Identify the path of the ship forms a right triangle with le

**Problem Statement:** You are asked to study and list the basic trigonometric identities, which are fundamental formulas used in trigonometry. These identities include Reciprocal,

1. Find two positive coterminal angles of $-135^\circ$. Coterminal angles differ by multiples of $360^\circ$.

1. **State the problem:** Calculate the value of the summation $$\sum_{n=3}^{2017} \sin \left( \frac{(n!)\pi}{36} \right)$$.

1. Stating the problem: Calculate the values of the following expressions involving trigonometric functions: a. $\sin 60^\circ \cos 45^\circ + \sec 30^\circ \cot 60^\circ$

1. Stating the problem: We need to find the principal solution for $\csc \theta = -6$. 2. Recall the definition: $\csc \theta = \frac{1}{\sin \theta}$, so $\csc \theta = -6$ means

1. Problem 21: Find the length of side AC in a right triangle ABC with \( \angle A = 90^\circ \), \( \angle B = 60^\circ \), and \( \angle C = 30^\circ \), given \( AB = 6 \) cm. 2

1. State the problem: Simplify the expression $$\cos\left(\frac{3}{2}\pi - 3x\right) + \sin(\pi - 5x)$$ and identify which option from A to E matches it. 2. Simplify each term usin