11B - Defining the circular functions
Sine and cosine:
The unit circle is used to define the sine and cosine functions, two of our circular functions. On the unit circle we can extent a line from the origin to the circumference to a point \(P\), this line makes an angle of \(\theta^c\):
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Tangent:
In addition to the sine and cosine functions, the tangent function can also be defined using the unit circle. A tangent is drawn to the unit circle at the point \((1,0)\). Recall that a tangent is a line that touches a curve only once.
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The four quadrants:
The unit circle can be split into four quadrants divided by the \(x\)- and \(y\)-axis:
11B - Example 1: Finding the values of sine, cosine and tangent using the unit circle
Determine the value of \(sin(\theta)\), \(cos(\theta)\) and \(tan(\theta)\) when
(a) \(\theta=\frac{\pi}{2}^c\) (b) \(\theta=\pi^c\) (c) \(\theta=\frac{3\pi}{2}^c\) (d) \(\theta=2\pi^c\) (e) \(\theta=\frac{7\pi}{2}^c\) (f) \(\theta=-\frac{\pi}{2}^c\) |
11B - Example 1: Video solution
11B - Example 1: Practice
Question 1: Find the value of (a) \(sin(\frac{5\pi}{2})\) (b) \(cos(\frac{5\pi}{2})\) (c) \(tan(\frac{5\pi}{2})\) Question 2: Find the value of (a) \(sin(-3\pi)\) (b) \(cos(-3\pi)\) (c) \(tan(-3\pi)\) 11B - Example 1: Solutions
Question 1: Find the value of (a) \(sin(\frac{5\pi}{2})=1\) (b) \(cos(\frac{5\pi}{2})=0\) (c) \(tan(\frac{5\pi}{2})=\)undefined Question 2: Find the value of (a) \(sin(-3\pi)=0\) (b) \(cos(-3\pi)=-1\) (c) \(tan(-3\pi)=0\) |
11B - Example 1: Finding the values of sine, cosine and tangent using symmetry properties
Given that \(sin(\theta)=0.891\), \(cos(\theta)=0.454\) and \(tan(\theta)=1.965\), where \(\theta\) is in the first quadrant. Using the symmetry properties of the unit circle, find the values of sine, cosine and tangent in the other three quadrants.
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11B - Example 2: Video solution
11B - Example 2: Practice
Question 1: ABC 11B - Example 2: Solutions
Question 1: ABC |