How to Know Sin Cos and Tan Without Calculator

Reference Angles and Common Values

The first step in calculating trigonometric values without a calculator is understanding reference angles. A reference angle is the smallest angle that a terminal side of a given angle makes with the x-axis. For any angle θ, the reference angle (θ') can be found using:

Common reference angles and their corresponding sine, cosine, and tangent values are shown in the table below:

For angles greater than 90°, you can use the reference angle to find the trigonometric values by considering the quadrant in which the angle lies. The sign of each trigonometric function depends on the quadrant:

• Quadrant I: All positive
• Quadrant II: sin positive, cos and tan negative
• Quadrant III: tan positive, sin and cos negative
• Quadrant IV: cos positive, sin and tan negative

Unit Circle Method

The unit circle is a circle with a radius of 1 centered at the origin (0,0) in the coordinate plane. Any angle θ drawn from the positive x-axis to a point (x, y) on the unit circle corresponds to the coordinates (cosθ, sinθ).

To find sinθ and cosθ using the unit circle:

1. Draw the angle θ from the positive x-axis.
2. Find the intersection point of the terminal side with the unit circle.
3. The x-coordinate of the point is cosθ.
4. The y-coordinate of the point is sinθ.

For example, to find sin(120°) and cos(120°):

1. Draw a 120° angle from the positive x-axis.
2. The terminal side intersects the unit circle at the point (-0.5, √3/2).
3. Therefore, cos(120°) = -0.5 and sin(120°) = √3/2.

Special Right Triangles

Special right triangles are triangles with specific angle measures that have sides in a consistent ratio. The two most common special right triangles are the 30-60-90 triangle and the 45-45-90 triangle.

30-60-90 Triangle

A 30-60-90 triangle has sides in the ratio 1 : √3 : 2. The sides opposite the 30°, 60°, and 90° angles are in the ratio 1 : √3 : 2.

Using this triangle, you can find the sine, cosine, and tangent of 30° and 60°:

• sin(30°) = opposite/hypotenuse = 1/2 = 0.5
• cos(30°) = adjacent/hypotenuse = √3/2 ≈ 0.866
• tan(30°) = opposite/adjacent = 1/√3 ≈ 0.577
• sin(60°) = opposite/hypotenuse = √3/2 ≈ 0.866
• cos(60°) = adjacent/hypotenuse = 1/2 = 0.5
• tan(60°) = opposite/adjacent = √3 ≈ 1.732

45-45-90 Triangle

A 45-45-90 triangle is an isosceles right triangle with two 45° angles and a 90° angle. The sides are in the ratio 1 : 1 : √2.

Using this triangle, you can find the sine, cosine, and tangent of 45°:

• sin(45°) = opposite/hypotenuse = 1/√2 ≈ 0.707
• cos(45°) = adjacent/hypotenuse = 1/√2 ≈ 0.707
• tan(45°) = opposite/adjacent = 1/1 = 1

Trigonometric Identities

Trigonometric identities are equations that relate trigonometric functions to each other. These identities can be used to simplify expressions and find values of trigonometric functions.

Pythagorean Identities

The Pythagorean identities relate the sine and cosine functions:

These identities can be used to find missing values when one trigonometric function is known.

Angle Sum and Difference Identities

The angle sum and difference identities allow you to find the sine, cosine, and tangent of the sum or difference of two angles:

These identities are useful for finding trigonometric values of angles that are sums or differences of known angles.

Example Calculations

Let's work through some examples to demonstrate how to find sine, cosine, and tangent values without a calculator.

Example 1: Finding sin(150°)

To find sin(150°), we can use the angle sum identity or reference angles.

1. First, find the reference angle: 150° - 180° = 30°.
2. 150° is in the second quadrant where sine is positive.
3. Therefore, sin(150°) = sin(30°) = 0.5.

Example 2: Finding cos(210°)

To find cos(210°), we can use the reference angle method.

1. Find the reference angle: 210° - 180° = 30°.
2. 210° is in the third quadrant where cosine is negative.
3. Therefore, cos(210°) = -cos(30°) ≈ -0.866.

Example 3: Finding tan(105°)

To find tan(105°), we can use the angle sum identity.

1. Express 105° as 60° + 45°.
2. Use the tangent addition formula: tan(A + B) = (tanA + tanB) / (1 - tanA tanB).
3. tan(105°) = (tan60° + tan45°) / (1 - tan60°tan45°) = (√3 + 1) / (1 - √3*1) ≈ (1.732 + 1) / (1 - 1.732) ≈ 2.732 / -0.732 ≈ -3.732.