Common Values of Sine, Cosine, & Tangent: Videos & Practice Problems

Trigonometric functions connect angles to points on the unit circle, with cosine and sine values represented as x and y coordinates. To memorize key angles (30°, 45°, 60°), use the 1, 2, 3 rule or the left hand rule. The 1, 2, 3 rule involves counting values clockwise and counterclockwise, while the left hand rule uses fingers to represent angles. Tangent values can be derived by dividing sine by cosine. Key equations include ${\tan }^{\theta }=\frac{\sin }{\cos }$.

Sine, Cosine, & Tangent of 30°, 45°, & 60°

Trigonometric functions establish a relationship between angles and their corresponding coordinates on the unit circle, where the x-coordinate represents the cosine value and the y-coordinate represents the sine value of that angle. To effectively solve trigonometric problems, it is essential to memorize the sine and cosine values for the three most common angles: 30°, 45°, and 60°. Two effective methods for memorization are the 1, 2, 3 rule and the left hand rule.

The 1, 2, 3 rule begins with the understanding that all sine and cosine values for these angles can be expressed as a fraction involving the square root of a number over 2. Starting with the angle of 60°, you count clockwise for the x-values (cosine) and counterclockwise for the y-values (sine). For example, the cosine of 60° is calculated as:

\[\cos(60°) = \frac{\sqrt{1}}{2} = \frac{1}{2}\]

Similarly, the sine of 30° is:

\[\sin(30°) = \frac{\sqrt{1}}{2} = \frac{1}{2}\]

To find the tangent of an angle, you can divide the sine by the cosine. For 30°, this results in:

\[\tan(30°) = \frac{\sin(30°)}{\cos(30°)} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\]

The left hand rule offers a more visual approach. By positioning your left hand with the pinky at 0° and the thumb at 90°, each finger represents a common angle: 30°, 45°, and 60°. To find the sine and cosine values, fold down the finger corresponding to the angle. For instance, for 30°, count the fingers above the folded finger for cosine and below for sine. Thus, for cosine:

\[\cos(30°) = \frac{\sqrt{3}}{2}\]

And for sine:

\[\sin(30°) = \frac{\sqrt{1}}{2} = \frac{1}{2}\]

Again, the tangent can be calculated as:

\[\tan(30°) = \frac{\sin(30°)}{\cos(30°)} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\]

Both methods provide a solid foundation for memorizing trigonometric values in the first quadrant, enabling you to tackle a variety of trigonometric problems with confidence.

Example 1

Understanding the unit circle is essential for mastering trigonometry, particularly when it comes to angles and their corresponding sine, cosine, and tangent values. The unit circle is a circle with a radius of 1, centered at the origin of a coordinate system, and it provides a visual representation of these trigonometric functions.

To begin filling in the first quadrant of the unit circle, start with the key angle measures. The angles in degrees are 0°, 30°, 45°, 60°, and 90°, which correspond to the following radian measures: 0, \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), and \(\frac{\pi}{2}\). These relationships can be derived by recognizing that:

• 0° is at the point (1, 0).
• 90° is at the point (0, 1).
• 30° is \(\frac{1}{3}\) of 90°, leading to the radian measure of \(\frac{\pi}{6}\).
• 45° is halfway between 0° and 90°, resulting in \(\frac{\pi}{4}\).
• 60° is double the 30° angle, giving \(\frac{\pi}{3}\).

Next, the sine and cosine values for these angles can be determined using the square root method. For angles in the first quadrant, the sine (y-coordinate) and cosine (x-coordinate) values can be expressed as:

• For 0°: \((1, 0)\) → \(\cos(0) = 1\), \(\sin(0) = 0\)
• For 30°: \((\frac{\sqrt{3}}{2}, \frac{1}{2})\) → \(\cos(30) = \frac{\sqrt{3}}{2}\), \(\sin(30) = \frac{1}{2}\)
• For 45°: \((\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})\) → \(\cos(45) = \frac{\sqrt{2}}{2}\), \(\sin(45) = \frac{\sqrt{2}}{2}\)
• For 60°: \((\frac{1}{2}, \frac{\sqrt{3}}{2})\) → \(\cos(60) = \frac{1}{2}\), \(\sin(60) = \frac{\sqrt{3}}{2}\)
• For 90°: \((0, 1)\) → \(\cos(90) = 0\), \(\sin(90) = 1\)

To find the tangent values, which are defined as the ratio of sine to cosine (or \(y/x\)), we can calculate:

• Tangent of 0°: \(\frac{0}{1} = 0\)
• Tangent of 30°: \(\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} \approx \frac{\sqrt{3}}{3}\) (after rationalizing the denominator)
• Tangent of 45°: \(\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1\)
• Tangent of 60°: \(\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}\)
• Tangent of 90°: \(\frac{1}{0}\) is undefined.

By practicing these calculations and familiarizing yourself with the unit circle, you can enhance your understanding of trigonometric functions and their applications. Repetition and reasoning through the relationships between angles, radians, and their corresponding sine, cosine, and tangent values will solidify your grasp of this fundamental concept in mathematics.

Here’s what students ask on this topic:

What are the sine, cosine, and tangent values for 30 degrees?

For 30 degrees, the sine, cosine, and tangent values are as follows:

$\sin (30°)=\frac{1}{2}$

$\cos (30°)=\frac{\sqrt{3}}{2}$

$\tan (30°)=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}$

How can I use the 1, 2, 3 rule to memorize trigonometric values?

The 1, 2, 3 rule helps you memorize trig values for 30°, 45°, and 60° by counting values on the unit circle. Start with the square root over 2. For cosine (x-values), count 1, 2, 3 clockwise starting from 60°. For sine (y-values), count 1, 2, 3 counterclockwise starting from 30°. Simplify the square roots: cos(60°) = 1/2, sin(30°) = 1/2, etc. For tangent, divide sine by cosine. For example, tan(30°) = sin(30°)/cos(30°) = (1/2)/(√3/2) = 1/√3 = √3/3.

What is the left hand rule for memorizing trigonometric values?

The left hand rule uses your left hand to represent the first quadrant of the unit circle. Your pinky is 0°, and your thumb is 90°. Fold the finger representing the angle (e.g., 30°, 45°, 60°). Count fingers above and below the folded finger. For cosine, use the square root of the number of fingers above divided by 2. For sine, use the square root of the number of fingers below divided by 2. For tangent, divide the square root of fingers below by the square root of fingers above. For example, for 30°, cos(30°) = √3/2, sin(30°) = 1/2, tan(30°) = 1/√3 = √3/3.

How do you find the tangent value of an angle using sine and cosine?

The tangent of an angle can be found by dividing the sine value by the cosine value. The formula is:

$\tan \theta =\frac{\sin }{\theta }$

For example, for 45°, sin(45°) = √2/2 and cos(45°) = √2/2. Therefore, tan(45°) = (√2/2) / (√2/2) = 1.

What are the sine, cosine, and tangent values for 45 degrees?

For 45 degrees, the sine, cosine, and tangent values are as follows:

$\sin (45°)=\frac{\sqrt{2}}{2}$

$\cos (45°)=\frac{\sqrt{2}}{2}$

$\tan (45°)=\frac{\sqrt{2}}{2}/\frac{\sqrt{2}}{2}=1$