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Trigonometric Ratios Of 30°, 45°, And 60°

Derive and use the trigonometric ratios for 30, 45, and 60 degrees using geometric properties.

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Deriving Ratios for 45°

Use an isosceles right triangle to find ratios for 45 degrees.

Setting up the 45° Right Triangle

To find the exact trigonometric values for $45^\circ$ , we can use geometry. Consider a right-angled triangle where one of the acute angles is exactly $45^\circ$ .

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Equilateral Triangle for 30° and 60°

Visual of an equilateral triangle cut in half to create a 30-60-90 triangle.

Clean scientific diagram of an equilateral triangle labeled ABC. A perpendicular line segment AD drops from top vertex A…

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Deriving Ratios for 30° and 60°

Use a bisected equilateral triangle to find ratios for 30 and 60 degrees.

The Equilateral Setup

To derive ratios for $30^\circ$ and $60^\circ$ , we start with an equilateral triangle where every angle is $60^\circ$ . To keep the math clean, let's assign every side a length of $2a$ .

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Key Values: 30°, 45°, 60°

Summary table of the exact values derived.

Standard Angle Values

30° Angle

sin:

\frac{1}{2}

cos:

\frac{3}{2}

tan:

\frac{1}{3}

45° Angle

sin:

\frac{1}{2}

cos:

\frac{1}{2}

tan:

1

60° Angle

sin:

\frac{3}{2}

cos:

\frac{1}{2}

tan:

3

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Example 6: Using Known Angles

Determine side lengths using a known 30 degree angle.

Problem.
In $\triangle ABC$ , right-angled at $B$ , the side $AB = 5\text{ cm}$ and $\angle ACB = 30^\circ$ . Determine the lengths of the sides $BC$ and $AC$ .

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Evaluate Expression with Specific Angles

Calculate 2 tan^2(45) + cos^2(30) - sin^2(60).

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Evaluate the expression: $2 \tan^2(45^\circ) + \cos^2(30^\circ) - \sin^2(60^\circ)$

← previousCalculating All Trigonometric Ratios next →Trigonometric Ratios Of 0° And 90°

Deriving Ratios for 45°

Use an isosceles right triangle to find ratios for 45 degrees.

Setting up the 45° Right Triangle

To find the exact trigonometric values for $45^\circ$ , we can use geometry. Consider a right-angled triangle where one of the acute angles is exactly $45^\circ$ .

1 / 4

Deriving Ratios for 30° and 60°

Use a bisected equilateral triangle to find ratios for 30 and 60 degrees.

The Equilateral Setup

To derive ratios for $30^\circ$ and $60^\circ$ , we start with an equilateral triangle where every angle is $60^\circ$ . To keep the math clean, let's assign every side a length of $2a$ .

1 / 5

Example 6: Using Known Angles

Determine side lengths using a known 30 degree angle.

Problem.
In $\triangle ABC$ , right-angled at $B$ , the side $AB = 5\text{ cm}$ and $\angle ACB = 30^\circ$ . Determine the lengths of the sides $BC$ and $AC$ .

1 / 5

Evaluate Expression with Specific Angles

Calculate 2 tan^2(45) + cos^2(30) - sin^2(60).

1 / 5

Evaluate the expression: $2 \tan^2(45^\circ) + \cos^2(30^\circ) - \sin^2(60^\circ)$