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Trig Identities (Part 1)
Definition
The cosine and sine functions are defined on the unit circle, x2 + y2 = 1. The independent variable of these functions, typically written θ, is any real number, with a unit called radians. As you can see in the picture below, the value of the cosine function at θ is the x-coordinate of the corresponding point on the unit circle. The value of the sine function at θ is the y-coordinate of the point. See this LiveMath web page for an animation.
Definition
The cosine and sine functions are defined on the unit circle, x2 + y2 = 1. The independent variable of these functions, typically written θ, is any real number, with a unit called radians. As you can see in the picture below, the value of the cosine function at θ is the x-coordinate of the corresponding point on the unit circle. The value of the sine function at θ is the y-coordinate of the point. See this LiveMath web page for an animation.
Contrary to popular misunderstanding, θ is allowed to take any real value between –∞ and ∞. Some people are obsessed with triangles, hence restricting the values of θ between 0 and π. What is defined with the unit circle does apply to triangles as a special case.
Never use degrees with θ. Use radians instead as a unit for θ when dealing with trigonometric functions in calculus. Why? Because if you used degrees, all of the differentiation & integration formulas with trigonometric functions break down.
Identities to Memorize (You have no choice. So, bite the bullet.)
| Names | Identities |
| Sum Identities | [1] sin(u + v)
= sin(u)cos(v)
+ cos(u)sin(v) [2] cos(u + v) = cos(u)cos(v) – sin(u)sin(v) |
| Difference Identities | [3] sin(u – v)
= sin(u)cos(v)
– cos(u)sin(v) [4] cos(u – v) = cos(u)cos(v) + sin(u)sin(v) |
| Pythagorean Identity | [5] cos2(θ) + sin2(θ) = 1 |
| Cosine is even | [6] cos(– u)
= cos(u) That is why the graph of cosine is symmetrical with respect to the vertical axis. |
| Sine is odd | [7] sin(– u)
= – sin(u) That is why the graph of sine is symmetrical with respect to the origin. |
The graph of the sine function, for example, is derived as in this LiveMath animation.
Continue to Part 2.
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Disclaimer • Updated September 4, 2008
Email: mitsuma@kutztown.edu
Phone: +1 (610) 462-WPJW
Phone: +1 (610) 462-WPJW