When drawn from the origin, a line 45 degrees above (counterclockwise from) the positive x-axis lies in quadrant I. Cosine is defined as the ratio between the adjacent side of a triangle and the hypotenuse of the triangle. A right triangle can be drawn in quadrant I composed of any point on that line, the origin and a point on the x-axis. The hypotenuse of this triangle is considered a length, and is therefore positive. The adjacent side of this triangle lies along the positive x-axis. Since the adjacent side and hypotenuse are both represented by positive numbers, the fraction A/H is positive. Therefore, cos 45 is positive.

When drawn from the origin, a line 45 degrees above (counterclockwise from) the positive x-axis lies in quadrant I. Cosine is defined as the ratio between the adjacent side of a triangle and the hypotenuse of the triangle. A right triangle can be drawn in quadrant I composed of any point on that line, the origin and a point on the x-axis. The hypotenuse of this triangle is considered a length, and is therefore positive. The adjacent side of this triangle lies along the positive x-axis. Since the adjacent side and hypotenuse are both represented by positive numbers, the fraction A/H is positive. Therefore, cos 45 is positive.

The domain of a function is the range of all possible inputs, or x-values, that yield a real value for f(x). Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Sine is defined as the ratio between the side length opposite to the angle in question and the hypotenuse (SOH, or sin x = opposite/hypotenuse). In any triangle created by the angle x and the x-axis, the hypotenuse is a nonzero number. As a result, the denominator of the fraction created by the definition sin x = opposite/hypotenuse is not equal to zero for any angle value x. Therefore, the domain of f(x) = sin x is all real numbers.

The domain of a function is the range of all possible inputs, or x-values, that yield a real value for f(x). Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Sine is defined as the ratio between the side length opposite to the angle in question and the hypotenuse (SOH, or sin x = opposite/hypotenuse). In any triangle created by the angle x and the x-axis, the hypotenuse is a nonzero number. As a result, the denominator of the fraction created by the definition sin x = opposite/hypotenuse is not equal to zero for any angle value x. Therefore, the domain of f(x) = sin x is all real numbers.

We begin with our original equation:

(Pythagorean Identity)

Looking at the unit circle we see that at and . We must plug these back into our original equation to validate them.

Checking

Checking

And so our only solution is

We begin with our original equation:

(Pythagorean Identity)

Looking at the unit circle we see that at and . We must plug these back into our original equation to validate them.

Checking

Checking

And so our only solution is

We begin with our original equation.

(Pythagorean Identity)

(Double-Angle Formula)

We know that will be equal to for when is any multiple of and when . We need to check both solutions (we will simply check for simplicity) to make sure they are valid solutions.

Checking :

Checking

By checking our solutions, we see the only solution to this equation is

When drawn from the origin, a line 135 degrees counterclockwise from the positive x-axis lies in quadrant II. Sine is defined as the ratio between the opposite side of a triangle and the hypotenuse of the triangle. A right triangle can be drawn in quadrant II composed of any point on that line, the origin and a point on the x-axis. The hypotenuse of this triangle is considered a length, and is therefore positive. The opposite side of this triangle lies above the x-axis, and is therefore represented by a positive number. Since the opposite side and hypotenuse are both represented by positive numbers, the fraction O/H is positive. Therefore, sin 135 is positive.

The domain of a function is the range of all possible inputs, or x-values, that yield a real value for f(x). Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Cosine is defined as the ratio between the side length opposite to the angle in question and the hypotenuse (CAH, or cos x = adjacent/hypotenuse). In any triangle created by the angle x and the x-axis, the hypotenuse is a nonzero number. As a result, the denominator of the fraction created by the definition cos x = adjacent/hypotenuse is not equal to zero for any angle value x. Therefore, the domain of f(x) = cos x is all real numbers.

The domain of a function is the range of all possible inputs, or x-values, that yield a real value for f(x). Trigonometric functions are equal to 0, 1, -1 or undefined when the angle lies on an axis, meaning that the angle is equal to 0, 90, 180 or 270 degrees (0, (pi)/2, pi or 3(pi)/2 in radians.) Trigonometric functions are undefined when they represent fractions with denominators equal to zero. Cosine is defined as the ratio between the side length opposite to the angle in question and the hypotenuse (CAH, or cos x = adjacent/hypotenuse). In any triangle created by the angle x and the x-axis, the hypotenuse is a nonzero number. As a result, the denominator of the fraction created by the definition cos x = adjacent/hypotenuse is not equal to zero for any angle value x. Therefore, the domain of f(x) = cos x is all real numbers.