📏 trigonometry

1. The problem is to sketch the graph of $y = \cos x$ for $0 \leq x \leq 360^\circ$. 2. The cosine function is defined as $y = \cos x$, where $x$ is the angle in degrees.

1. The problem is to sketch the graph of the function $y = \sin x$ for $0 \leq x \leq 360^\circ$. 2. The sine function is periodic with period $360^\circ$, and it oscillates betwee

1. **State the problem:** Prove the trigonometric identity $$\sin x \tan x + \cos x = \sec x$$. 2. **Recall the definitions and formulas:**

1. **State the problem:** We are given a triangle with sides 11.4 m and 6.0 m, and an angle of 31° opposite the 11.4 m side. We need to find the angle $\alpha$ opposite the 6.0 m s

1. **Problem statement:** Simplify the expression $$\frac{\cos^2\left(\frac{\pi}{2} - x\right)}{\cos x}$$. 2. **Recall the co-function identity:** $$\cos\left(\frac{\pi}{2} - x\rig

1. **Rewrite** $\sqrt{x^2 + 36}$ using the substitution $x = 6 \tan \theta$, where $0 < \theta < \frac{\pi}{2}$. 2. Substitute $x = 6 \tan \theta$ into the expression:

1. **Problem:** Simplify the expression $$\sec^2 x - 1$$ and identify which function it equals. 2. **Recall the Pythagorean identity:** $$\sec^2 x = 1 + \tan^2 x$$.

1. **Problem a:** Simplify $\cot x \sec x$. Recall the definitions:

1. **Simplify** $\frac{1 + \sec(-\theta)}{\sin(-\theta) + \tan(-\theta)}$. Recall the even-odd properties of trig functions:

1. **State the problem:** Solve the trigonometric equation $4\sin x + 3\cos x = 2$ for $x$. 2. **Formula and approach:** We use the method of expressing $a\sin x + b\cos x$ as $R\s

1. The problem asks to find $\cot \theta$ given that $\sec \theta = 2 \times \frac{\sqrt{3}}{3}$.\n\n2. Recall the identity relating $\sec \theta$ and $\cos \theta$: $$\sec \theta

1. **State the problem:** Simplify the expression $$\frac{\sec x + \sin x \sec x}{\cos x}$$. 2. **Recall definitions and formulas:**

1. The problem asks to find the vertical shift of the trigonometric function $$f(x) = 3 \cos(x) - 3$$. 2. The general form of a cosine function with vertical shift is $$f(x) = A \c

1. **State the problem:** We have a right triangle formed by a tent pole (vertical side) of length 26 ft, a base (horizontal side) of length 20 ft, and the hypotenuse representing

1. **State the problem:** We have an observatory 150 feet high and an angle of depression of 25° to an island. We want to find the horizontal distance from the observatory to the i

1. Problem: Rozwiąż trójkąt, w którym dane są kąty $\alpha=45^\circ$, $\beta=60^\circ$ oraz bok $c=6$. 2. Wzór: Wykorzystamy twierdzenie sinusów, które mówi, że w dowolnym trójkąci

1. **Problem statement:** Sally is located such that from checkpoint A she is N32.6°E, and from checkpoint B she is N15.0°W. The distance between checkpoints A and B is 45 km, with

1. **State the problem:** Solve the trigonometric equation $$3 + 5 \sin \theta = 1$$ for $$0^\circ \leq \theta \leq 360^\circ$$. 2. **Isolate the sine term:**

1. The problem involves a triangle with sides 90 cm, 130 cm, and a side labeled $e$. Angles opposite these sides are $f$ and $g$. We want to find equations for $e$, $f$, and $g$ us

1. **State the problem:** Solve the trigonometric equation $$3 + 5 \sin \theta = 1$$ for $$\theta$$ in the interval $$0^\circ \leq \theta \leq 360^\circ$$. 2. **Isolate $$\sin \the

1. Let's start by understanding the problem: We need to solve the equations for $0 \leq \theta \leq 2\pi$. 2. The first equation is $\sin 3\theta = \frac{\sqrt{3}}{2}$.