Trigonometry Fundamentals

In a right-angled triangle, the sine of an angle is defined as the ratio of the ______ to the hypotenuse.

The trigonometric identity that relates the square of sine and cosine of an angle to 1 is known as the ______ identity.

The ______ function is periodic with a period of π.

The reciprocal of the sine function is the ______ function.

To find the angle whose tangent is a given value, one would use the ______ function.

The formula for the cosine of the sum of two angles, cos(A + B), involves both cosines and sines of A and B and is given by cos(A)cos(B) ______ sin(A)sin(B).

In the unit circle, the x-coordinate of a point corresponding to an angle θ represents the ______ of θ.

The identity that expresses $\tan(2\theta)$ in terms of $\tan(\theta)$ is $\tan(2\theta) = (2\tan(\theta)) / (1 - \tan^2(\theta))$, which is known as the ______ identity.

For any angle θ, the value of $\sin^2(\theta) + \cos^2(\theta)$ is always equal to ______.

The ______ function is undefined when x is an integer multiple of π.

In fields such as astronomy and navigation, trigonometry's main purpose is determining ______ and distances.

The ______ of an angle is the ratio of the adjacent side to the hypotenuse in a right-angled triangle.

When dealing with wave motion, oscillations, or projectile motion, ______ is applied in physics to analyze these phenomena.

The ______ of the sine function is all real numbers, while its range is [-1, 1].

For an angle θ, $\cot(\theta)$ can be expressed as $\cos(\theta) / \sin(\theta)$, illustrating the relationship between trigonometric ______.

The ______ is used to define trigonometric functions for all real numbers, not just angles in a right-angled triangle.

The ______ of the tangent function consists of all real numbers except where $x = (π/2) + nπ$, where n is an integer.

If $\sin(y) = x$, then y is the ______ of x, denoted as $\arcsin(x)$ or $\sin^{-1}(x)$.

In surveying, trigonometry helps measure land areas and ______.

Using the appropriate 'half angle identity', $\sin(\theta/2)$ can be expressed with the equation $\pm\sqrt{((1 - \cos(\theta))/2)}$, and this contains both a positive and ______ radical.