Common Values of Sine, Cosine, and Tangent for 30°, 45°, and 60°

Common Values of Sine, Cosine, and Tangent

Special Angles: 30°, 45°, and 60°

In precalculus, the trigonometric values for the angles 30°, 45°, and 60° are frequently used in problem solving. These values can be memorized using geometric reasoning or mnemonic devices. Understanding these values is essential for solving trigonometric equations and for applications in geometry and calculus.

• Sine, cosine, and tangent are fundamental trigonometric functions defined for a given angle in a right triangle or on the unit circle.

• The values for 30°, 45°, and 60° are especially important and recur in many mathematical contexts.

The 1-2-3 Rule

This rule uses the unit circle and right triangle geometry to derive the values of sine and cosine for the special angles.

• For angles 30°, 45°, and 60°, the coordinates on the unit circle can be used to find sine and cosine:

Angle	$heta$ (degrees)	$heta$ (radians)	$ext{sin}( heta)$	$ext{cos}( heta)$	$ext{tan}( heta)$
30°	30	$rac{ ext{π}}{6}$	$rac{1}{2}$	$rac{ ext{√3}}{2}$	$rac{1}{ ext{√3}}$
45°	45	$rac{ ext{π}}{4}$	$rac{1}{ ext{√2}}$	$rac{1}{ ext{√2}}$	$1$
60°	60	$rac{ ext{π}}{3}$	$rac{ ext{√3}}{2}$	$rac{1}{2}$	$ext{√3}$

• These values can be visualized on the unit circle, where the x-coordinate gives the cosine and the y-coordinate gives the sine of the angle.

The Left Hand Rule

This mnemonic uses your left hand to quickly recall the sine and cosine values for 0°, 30°, 45°, 60°, and 90°.

• Hold up your left hand and assign each finger to an angle: thumb (0°), index (30°), middle (45°), ring (60°), pinky (90°).

• To find $ext{sin}( heta)$, count the number of fingers to the left of the chosen finger, take the square root, and divide by 2:

• $ext{sin}( heta) = rac{ ext{√(number of fingers left of finger)}}{2}$

• To find $ext{cos}( heta)$, count the number of fingers to the right of the chosen finger, take the square root, and divide by 2:

• $ext{cos}( heta) = rac{ ext{√(number of fingers right of finger)}}{2}$

Formulas and Examples

• Sine values: $ext{sin}~30^ ext{o} = rac{1}{2}$ $ext{sin}~45^ ext{o} = rac{1}{ ext{√2}}$ $ext{sin}~60^ ext{o} = rac{ ext{√3}}{2}$

• Cosine values: $ext{cos}~30^ ext{o} = rac{ ext{√3}}{2}$ $ext{cos}~45^ ext{o} = rac{1}{ ext{√2}}$ $ext{cos}~60^ ext{o} = rac{1}{2}$

• Tangent values: $ext{tan}~30^ ext{o} = rac{1}{ ext{√3}}$ $ext{tan}~45^ ext{o} = 1$ $ext{tan}~60^ ext{o} = ext{√3}$

Example Application

Suppose you are asked to find the sine, cosine, and tangent of 45°:

• $ext{sin}~45^ ext{o} = rac{1}{ ext{√2}}$

• $ext{cos}~45^ ext{o} = rac{1}{ ext{√2}}$

• $ext{tan}~45^ ext{o} = 1$

These values are used in solving right triangles, evaluating trigonometric expressions, and in calculus for limits and derivatives involving trigonometric functions.

Summary Table

Angle	$ext{sin}( heta)$	$ext{cos}( heta)$	$ext{tan}( heta)$
30°	$rac{1}{2}$	$rac{ ext{√3}}{2}$	$rac{1}{ ext{√3}}$
45°	$rac{1}{ ext{√2}}$	$rac{1}{ ext{√2}}$	$1$
60°	$rac{ ext{√3}}{2}$	$rac{1}{2}$	$ext{√3}$

Additional info: The notes also include a fill-in-the-blank exercise to reinforce memorization of these values using the unit circle diagram.