Special Right Triangles and Exact Values

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Evaluate exact values: $$\sin 45^{\circ} = ? \quad \text{and} \quad \cos 45^{\circ} = ?$$

$$\sin 45^{\circ} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$$ $$\cos 45^{\circ} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$$

Evaluate exact values: $$\tan 45^{\circ} = ? \quad \text{and} \quad \cot 45^{\circ} = ?$$

$$\tan 45^{\circ} = 1$$ $$\cot 45^{\circ} = 1$$

Evaluate exact values: $$\sin 30^{\circ} = ? \quad \text{and} \quad \cos 30^{\circ} = ?$$

$$\sin 30^{\circ} = \frac{1}{2}$$ $$\cos 30^{\circ} = \frac{\sqrt{3}}{2}$$

Evaluate exact values: $$\sin 60^{\circ} = ? \quad \text{and} \quad \cos 60^{\circ} = ?$$

$$\sin 60^{\circ} = \frac{\sqrt{3}}{2}$$ $$\cos 60^{\circ} = \frac{1}{2}$$

Evaluate exact values: $$\tan 30^{\circ} = ? \quad \text{and} \quad \tan 60^{\circ} = ?$$

$$\tan 30^{\circ} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$$ $$\tan 60^{\circ} = \sqrt{3}$$

Evaluate exact values: $$\sec 60^{\circ} = ? \quad \text{and} \quad \csc 30^{\circ} = ?$$

$$\sec 60^{\circ} = \frac{1}{\cos 60^{\circ}} = \frac{1}{1/2} = 2$$ $$\csc 30^{\circ} = \frac{1}{\sin 30^{\circ}} = \frac{1}{1/2} = 2$$

Evaluate exact value: $$\sec 45^{\circ} = ?$$

$$\sec 45^{\circ} = \frac{1}{\cos 45^{\circ}} = \frac{2}{\sqrt{2}} = \sqrt{2}$$

Evaluate exact value: $$\csc 60^{\circ} = ?$$

$$\csc 60^{\circ} = \frac{1}{\sin 60^{\circ}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$$

In a $30^{\circ}$-$60^{\circ}$-$90^{\circ}$ triangle, the hypotenuse is $10$. What are the lengths of the shorter leg and longer leg?

Shorter leg (opposite $30^{\circ}$): $$\frac{10}{2} = 5$$ Longer leg (opposite $60^{\circ}$): $$5\sqrt{3}$$

In a $45^{\circ}$-$45^{\circ}$-$90^{\circ}$ triangle, each leg has length $7$. What is the hypotenuse length?

$$\text{Hypotenuse} = 7\sqrt{2}$$

How can an equilateral triangle of side length $2$ be used to derive the trig ratios for $30^{\circ}$ and $60^{\circ}$?

An equilateral triangle with side lengths of 2 bisected by an altitude of square root 3, forming two 30-60-90 right triangles with base lengths of 1 to derive standard trigonometric ratios.

Dropping an altitude splits the equilateral triangle into two congruent $30^{\circ}$-$60^{\circ}$-$90^{\circ}$ right triangles. The hypotenuse is $2$, the base leg is $1$, and the altitude leg is $\sqrt{2^2 - 1^2} = \sqrt{3}$.

Evaluate: $$(\sin 60^{\circ})^2 + (\cos 60^{\circ})^2 = ?$$

$$\left(\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{1}{2}\right)^2 = \frac{3}{4} + \frac{1}{4} = 1$$

What is the ratio of side lengths in a $45^{\circ}$-$45^{\circ}$-$90^{\circ}$ triangle?

A 45-45-90 special right triangle with two equal legs of length x opposite the 45-degree angles and a hypotenuse of length x times the square root of 2 opposite the 90-degree right angle.

$$1 : 1 : \sqrt{2}$$ Opposite sides to $45^{\circ}$ are equal legs $x$, and the hypotenuse is $x\sqrt{2}$.

What is the ratio of side lengths in a $30^{\circ}$-$60^{\circ}$-$90^{\circ}$ triangle?

Diagram of a 30-60-90 right triangle showing side lengths opposite the 30-degree angle as x, opposite the 60-degree angle as x times the square root of 3, and the hypotenuse as 2x.

$$1 : \sqrt{3} : 2$$ Side opposite $30^{\circ}$ is $x$, side opposite $60^{\circ}$ is $x\sqrt{3}$, and the hypotenuse is $2x$.

Special Right Triangles and Exact Values

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