📏 trigonometry

1. **State the problem:** Solve the equation $\cos \theta + 1 = 0$ for all values of $\theta$ where $\theta$ is in radians. 2. **Rewrite the equation:**

1. The problem is to identify the correct Pythagorean identities from the given options. 2. The fundamental Pythagorean identity is:

1. Problem statement: P is 12 km due north of Q, the bearing of R from P is 135^\circ and from Q is 120^\circ; find (i) angle PRQ and (ii) distance PR. 2. Set a coordinate frame: p

1. The problem is to sketch the graph of the function $y = \cos x$ for $0 \leq x \leq 360^\circ$.\n\n2. The cosine function is periodic with period $360^\circ$, and it oscillates b

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