Physics
It can be very help to memorize the values of sine and cosine for the most common angles (0, 30, 45, 60, and 90 degrees). For sine, as an angle increases from 0 degrees to 90 degrees, the value of sine also increases. The sine values for these angles are: sine of 0 is 0, sine of 30 is √1/2 (1/2), sine of 45 is √2/2, sine of 60 is √3/2, and sine of 90 is √4/2 (1). On the other hand, the values of cosine decrease as the angle size goes from 0 to 90 degrees. The cosine values for these angles are: cosine of 0 is √4/2 (1), cosine of 30 is √3/2, cosine of 45 is √2/2, cosine of 60 is √1/2 (1/2), and cosine of 90 is 0.
Lesson Outline
<ul> <li>Most common angles: 0, 30, 45, 60, and 90 degrees</li> <li>Starting with sine:</li> <ul> <li>As angle increases from 0 to 90 degrees, value of sine increases</li> <li>Sine values:</li> <ul> <li>Sine of 0° = 0 (√0/2)</li> <li>Sine of 30° = 1/2 (√1/2)</li> <li>Sine of 45° = √2/2</li> <li>Sine of 60° = √3/2</li> <li>Sine of 90° = 1 (√4/2)</li> </ul> </ul> <li>Moving to cosine:</li> <ul> <li>As angle size goes from 0 to 90 degrees, values of cosine decrease</li> <li>Cosine values:</li> <ul> <li>Cosine of 0° = 1 (√4/2)</li> <li>Cosine of 30° = √3/2</li> <li>Cosine of 45° = √2/2 (equal to sine of 45°)</li> <li>Cosine of 60° = 1/2 (√1/2)</li> <li>Cosine of 90° = 0 (√0/2)</li> </ul> </ul> <li>Recap: Sine values go up, cosine values go down</li> </ul>
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FAQs
Students should be familiar with sine and cosine values for standard angles such as 0°, 30°, 45°, 60°, and 90°. Common values include sin(0)=0, sin(30)=1/2, sin(45)=√2/2, sin(60)=√3/2, sin(90)=1, and for cosines: cos(0)=1, cos(30)=√3/2, cos(45)=√2/2, cos(60)=1/2, cos(90)=0.
The sine and cosine functions are related through the Pythagorean identity: sin^2(x) + cos^2(x) = 1. This identity demonstrates that while the sine of an angle gives the ratio of the opposite side to the hypotenuse, the cosine gives the ratio of the adjacent side to the hypotenuse. Both sine and cosine functions are periodic, with a period of 360° or 2π radians.
Several patterns can be seen in sine and cosine values for different angles. For every 90°, the values of sine and cosine are repeated or switched. For example, sine and cosine values for 30°, 150°, 210°, and 330° can be deduced from the values for 30° due to this pattern. Additionally, sine values correspond to the cosine values for an angle's complement, while the cosine values correspond to sine values for the angle's complement. This relationship is known as the cofunction identity.
