📏 trigonometry

1. **State the problem:** Solve the equation $2\cos^2 x = 1$ for $0 < x < 360$ degrees. 2. **Rewrite the equation:** Divide both sides by 2 to isolate $\cos^2 x$:

1. **Problem 5:** Given $\cos \theta = -\frac{3}{4}$ and $\theta$ is in quadrant II, find the other five trigonometric ratios. 2. **Step 1: Understand the quadrant and sign rules.*

1. **Problem Statement:** You are given three islands A, B, and C with bearings and distances from A:

1. **Problem statement:** In triangle DEF, right angled at E, with side DE = 50 cm and angle DEF = 17°, find the length of DF. 2. **Formula and rules:** In a right triangle, the si

1. **State the problem:** Solve the equation $$4(\tan x - 1) = 3(5 - 2 \tan x)$$ for $$0 < x < 360$$ degrees. 2. **Write the equation:** $$4\tan x - 4 = 15 - 6\tan x$$

1. **State the problem:** We need to find all angles $x$ such that $0^\circ < x < 360^\circ$ and $\tan x = -2$. 2. **Recall the tangent function properties:** The tangent function

1. **Problem statement:** Pierre is on a viewing deck 300 m above the ground. He looks down at point A with an angle of depression of 40° and then further down at point B with an a

1. **Problem Statement:** We need to sketch the graph of the function $$y=3\sin\left(x^2\right)$$ for the domain $$0 \leq x \leq 2\pi$$.

1. The problem is to understand how we determine that an angle is not negative. 2. Angles are typically measured from a reference line, usually the positive x-axis, in a counterclo

1. **Problem:** Determine the period, amplitude, phase shift, and horizontal shift of the function $f(x) = 4 \sin(4x - 3\pi)$. 2. **Formula and rules:** For a function of the form

1. The problem is to find the exact value of $\sin(\frac{\pi}{6})$. 2. The formula for sine of special angles is based on the unit circle and known values: $\sin(\frac{\pi}{6}) = \

1. The problem is to find the values of $\sin 60^\circ$, $\cos 60^\circ$, and $\tan 60^\circ$. 2. Recall the special angles in trigonometry: for $60^\circ$, the values are derived

1. The problem is to find the values of $\sin 45^\circ$, $\cos 45^\circ$, and $\tan 45^\circ$. 2. Recall the definitions and important values for these trigonometric functions at $

1. The problem asks to find $\sin 30^\circ$, $\cos 30^\circ$, and $\tan 30^\circ$. 2. Recall the definitions and values of sine, cosine, and tangent for special angles. For $30^\ci

1. **Problem statement:** We have a right triangle with a hypotenuse of length 12 and one angle of 30°. We need to find the length of side $a$, which is opposite the 30° angle. 2.

1. **Problem Statement:** Simplify the expression $\cot \alpha - \cot (\alpha + \beta)$. 2. **Recall the cotangent subtraction formula:** For any angles $x$ and $y$,

1. The problem asks to find the expression equivalent to $1 + \tan^2(A)$. 2. We use the Pythagorean identity in trigonometry:

1. We are asked to solve the equation $\sin(45^\circ - a) - 3\cos(45^\circ + a) + 1 = 0$ for $a$. 2. Recall the angle sum and difference formulas:

1. **State the problem:** We need to find the length of side JL in a right triangle JLK where angle L is 90°, angle K is 52°, and side KL is adjacent to angle K. We are given two c

1. **State the problem:** A farmer wants to build a fence around a right-angled triangular field. One angle is 53° and the side opposite this angle is 126 m. We need to find the to

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