Trigonometric Functions of a General Angle
Solved Problems
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Example 1
Find the value of the expression \[\sin \frac{\pi }{4}\cos \frac{\pi }{6}\tan \frac{\pi }{3}.\]
Example 2
Find the value of the expression \[\cot \frac{\pi }{6}\sec \frac{\pi }{3}\csc \frac{\pi }{4}\tan \frac{\pi }{6}.\]
Example 3
Calculate the values of the six trigonometric functions of α = π/6.
Example 4
Calculate the values of the six trigonometric functions of α = π/4.
Example 5
The terminal side of an angle \(\alpha\) in standard position contains the point \(P\left( { - 2,3} \right).\) Find the six trigonometric functions of the angle \(\alpha.\)
Example 6
The terminal side of an angle \(\beta\) in standard position contains the point \(Q\left( { - 8,-6} \right).\) Find the six trigonometric functions of the angle \(\beta.\)
Example 7
Calculate the sum of the series \[1 + \sin \frac{\pi }{6} + {\sin ^2}\frac{\pi }{6} + {\sin ^3}\frac{\pi }{6} + \ldots \]
Example 8
Calculate the sum of the series \[1 - \cos \frac{\pi }{4} + {\cos ^2}\frac{\pi }{4} - {\cos ^3}\frac{\pi }{4} + \ldots \]
Example 1.
Find the value of the expression \[\sin \frac{\pi }{4}\cos \frac{\pi }{6}\tan \frac{\pi }{3}.\]
Solution.
This expression contains trig functions of special angles. The values of these functions are given in the table above:
Hence
Example 2.
Find the value of the expression \[\cot \frac{\pi }{6}\sec \frac{\pi }{3}\csc \frac{\pi }{4}\tan \frac{\pi }{6}.\]
Solution.
Using the table above, we find that
Substitute these values into our expression:
Example 3.
Calculate the values of the six trigonometric functions of \(\alpha = \frac{\pi }{6}.\)
Solution.
The right triangle \(OAM\) is a special \(30\text{-}60\text{-}90\) triangle in which the hypotenuse is twice the length of the shorter leg. Therefore, we have
The cosine of \(\alpha = \frac{\pi }{6}\) can be found by the Pythagorean trig identity:
The other trigonometric functions of \(\alpha = \frac{\pi }{6}\) are given by
The table above with the values of trig functions for special angles is composed on the basis of these calculations.
Example 4.
Calculate the values of the six trigonometric functions of \(\alpha = \frac{\pi }{4}.\)
Solution.
We deal here with a \(45\text{-}45\text{-}90\) triangle. This is a right isosceles triangle, so it has equal legs. Suppose the length of a leg be \(x.\) By the Pythagorean theorem,
Since \(r = 1,\) we have
Compute the values of the other trig functions:
Example 5.
The terminal side of an angle \(\alpha\) in standard position contains the point \(P\left( { - 2,3} \right).\) Find the six trigonometric functions of the angle \(\alpha.\)
Solution.
Here \(x = -2,\) \(y = 3.\) The distance of the point \(P\left( { - 2,3} \right)\) from the origin is equal to
The trig functions of the angle \(\alpha\) are given by
Example 6.
The terminal side of an angle \(\beta\) in standard position contains the point \(Q\left( { - 8,-6} \right).\) Find the six trigonometric functions of the angle \(\beta.\)
Solution.
Determine the distance \(r\) from the origin to the point \(Q\left( { - 8,-6} \right):\)
Calculate the values of the trig functions:
Example 7.
Calculate the sum of the series \[1 + \sin \frac{\pi }{6} + {\sin ^2}\frac{\pi }{6} + {\sin ^3}\frac{\pi }{6} + \ldots \]
Solution.
Since \(\sin \frac{\pi }{6} = \frac{1}{2},\) we can write this expression in the form:
We have here an infinite geometric series with the initial term \(a_1 =1\) and the common ratio \(q = \frac{1}{2}.\) The sum of the geometric series is given by
Example 8.
Calculate the sum of the series \[1 - \cos \frac{\pi }{4} + {\cos ^2}\frac{\pi }{4} - {\cos ^3}\frac{\pi }{4} + \ldots \]
Solution.
We have here an infinite geometric series with the initial term \(a_1 = 1\) and the negative common ratio \(q = - \cos \frac{\pi }{4} = - \frac{{\sqrt 2 }}{2}.\) Determine the sum of the series: