How do the exact values of $\sin(45^\circ)$ and $\cos(45^\circ)$ compare?

They are identical, both equaling $\frac{1}{\sqrt{2}}$ (or $\frac{\sqrt{2}}{2}$). This is because a $45^\circ$ right-angled triangle is isosceles, meaning the opposite and adjacent sides are equal.

When comparing $\tan(30^\circ)$ and $\tan(60^\circ)$, which value is larger and why?

$\tan(60^\circ) = \sqrt{3}$ is larger than $\tan(30^\circ) = \frac{1}{\sqrt{3}}$. As the angle increases toward $90^\circ$, the opposite side grows relative to the adjacent side, increasing the tangent ratio.

What is the relationship between $\sin(\theta)$ and $\cos(90^\circ - \theta)$ regarding exact values?

They are equal. For example, $\sin(30^\circ) = \cos(60^\circ) = 0.5$. This is due to the co-function identity where the opposite side for one acute angle is the adjacent side for the other.

What is a common mistake when calculating $\tan(90^\circ)$ or $\tan(\frac{\pi}{2})$?

Students often try to assign it a numerical value like $0$ or $1$, but it is actually undefined. This happens because $\cos(90^\circ) = 0$, and the tangent ratio $\frac{\sin}{\cos}$ results in division by zero.

Why is it an error to write $\sin(60^\circ) = 0.866$ in a pure math proof?

$0.866$ is a rounded decimal approximation, not an exact value. In pure mathematics, you must use $\frac{\sqrt{3}}{2}$ to maintain precision and allow for further algebraic simplification.

What error occurs if you confuse the sides of the $30-60-90$ triangle?

You will swap the values of sine and cosine. Remember that $30^\circ$ is the smaller angle, so it must be opposite the shortest side ($1$), while $60^\circ$ is opposite the longer leg ($\sqrt{3}$).

Define 'Exact Value' in the context of trigonometry.

An exact value is a non-rounded representation of a trigonometric ratio, typically expressed as an integer, a fraction, or a surd (e.g., $\frac{1}{2}$ or $\sqrt{3}$).

What is the exact value of $\cos(\pi)$?

The exact value is $-1$. On the unit circle, $\pi$ radians ($180^\circ$) corresponds to the coordinate $(-1, 0)$, and cosine is the x-coordinate.

State the exact values for $\sin, \cos,$ and $\tan$ at $0^\circ$.

$\sin(0^\circ) = 0$, $\cos(0^\circ) = 1$, and $\tan(0^\circ) = 0$.

How can you derive the exact value of $\tan(45^\circ)$ without a calculator?

Use an isosceles right triangle where both legs are length $1$. Since $\tan = \frac{\text{Opposite}}{\text{Adjacent}}$, the ratio is $\frac{1}{1} = 1$.

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Trigonometry Exact Values

Summary

Trigonometry exact values are specific outputs of trigonometric functions for standard angles that can be expressed as rational numbers or surds rather than infinite decimals. These values are derived from the geometric properties of special triangles and the unit circle, providing a foundation for solving complex equations without rounding errors.

1. Definition & Core Concepts

2. Geometric Derivation: Special Triangles

Geometric derivation of exact values using a 45-45-90 triangle and a 30-60-90 triangle.

3. Boundary Values and the Unit Circle

4. Key Distinctions: Sine vs. Cosine Patterns

5. Exam Strategy & Tips

Trigonometry Exact Values

Summary

1. Definition & Core Concepts

Exact values refer to the precise mathematical representation of trigonometric ratios (sine, cosine, and tangent) for specific angles like $0^\circ$ , $30^\circ$ , $45^\circ$ , $60^\circ$ , and $90^\circ$ .

Unlike calculator outputs which often provide rounded decimal approximations, exact values use surds (roots) and fractions to maintain absolute precision in mathematical proofs and calculations.

These values are universal across both degrees and radians, where $30^\circ = \frac{\pi}{6}$ , $45^\circ = \frac{\pi}{4}$ , and $60^\circ = \frac{\pi}{3}$ .

2. Geometric Derivation: Special Triangles

The values for $45^\circ$ are derived from an isosceles right-angled triangle with side lengths of $1, 1,$ and a hypotenuse of $\sqrt{2}$ calculated via Pythagoras' Theorem.

The values for $30^\circ$ and $60^\circ$ are derived from an equilateral triangle with side length $2$ . By bisecting the triangle, we create a right-angled triangle with sides $1, \sqrt{3},$ and hypotenuse $2$ .

By applying SOH CAH TOA to these triangles, the ratios emerge naturally: for instance, $\sin(45^\circ) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{1}{\sqrt{2}}$ .

Geometric derivation of exact values using a 45-45-90 triangle and a 30-60-90 triangle.

3. Boundary Values and the Unit Circle

Angles at the boundaries of quadrants ( $0^\circ, 90^\circ, 180^\circ, 270^\circ, 360^\circ$ ) are best understood using the unit circle where the coordinates $(x, y)$ represent $(\cos \theta, \sin \theta)$ .

At $0^\circ$ , the coordinate is $(1, 0)$ , meaning $\cos(0) = 1$ and $\sin(0) = 0$ . At $90^\circ$ ( $\frac{\pi}{2}$ ), the coordinate is $(0, 1)$ , meaning $\cos(90^\circ) = 0$ and $\sin(90^\circ) = 1$ .

The tangent function is defined as $\frac{\sin \theta}{\cos \theta}$ , which explains why $\tan(90^\circ)$ is undefined (division by zero).