7.09 Using radians
Calculate the following trigonometric ratios to two decimal places:
\sin \dfrac{35 \pi}{16}
\cos 6.87
\sin 7.26
\tan 7.26
\tan \left(\dfrac{- 3 \pi}{7}\right)
\cos \dfrac{2 \pi}{3}
\sin \left( - \dfrac{4 \pi}{3}\right)
Consider the following diagram:
Find the length of side h.
Hence, state the exact value of:
Consider the following diagram:
Find the length of the hypotenuse, h.
Hence, state the exact value of:
Consider the diagram of the unit circle:
Find the exact value of:
\text{cosec } \dfrac{\pi}{4}
\cot \dfrac{\pi}{3}
Consider the unit circle diagram and state the exact value of the following trigonometric ratios:
\sin \dfrac{\pi}{2}
\cos \dfrac{3\pi}{2}
\tan \pi
\cos 0
\sec \dfrac{\pi}{2}
\sec \pi
\text{cosec } \dfrac{\pi}{2}
\sin \left(-2\pi\right)
Find the exact value of the following:
\sin \dfrac{\pi}{3} + \cos \dfrac{\pi}{3}
\sin \dfrac{\pi}{6} \cos \dfrac{\pi}{4}
\dfrac{\sin \dfrac{\pi}{3}}{\cos \dfrac{\pi}{6}}
\sin \dfrac{\pi}{4} \cos \dfrac{\pi}{6} + \tan \dfrac{\pi}{4}
\sin ^{2}\left(\dfrac{\pi}{6}\right) - \cos ^{2}\left(\dfrac{\pi}{3}\right)
2\sin ^{2}\left(\dfrac{\pi}{2}\right) + 3\cos ^{2}\left(\dfrac{\pi}{2}\right)
Consider the unit circle shown, where points A and B have the same \\ y-coordinates.
Suppose that \theta = \dfrac{10 \pi}{11}. State the size of the reference angle, \alpha.
Consider the unit circle shown, where the line through A and B passes through the origin, O.
Suppose that \theta = \dfrac{8 \pi}{7}. State the size of the reference angle, \alpha.
Consider the unit circle shown, where the points A and B have the same \\ x-coordinate.
Suppose that \theta = \dfrac{9 \pi}{5}. State the size of the reference angle, \alpha.
Find the exact value of the following:
\sin \dfrac{5 \pi}{6}
\tan \dfrac{3 \pi}{4}
\sin \dfrac{7 \pi}{6}
\cos \dfrac{7 \pi}{6}
\sin \dfrac{5 \pi}{3}
\cos \dfrac{5 \pi}{3}
\cos 4 \pi
\tan 9 \pi
\sin \dfrac{ 5\pi}{2}
\cos \dfrac{ 7\pi}{2}
\cos \dfrac{3 \pi}{4}
\tan \dfrac{7 \pi}{6}
\tan \dfrac{11 \pi}{6}
Find the exact value of the following:
\sin \left( - \dfrac{17 \pi}{6} \right)
\cos \left( - \dfrac{17 \pi}{6} \right)
\cos \left( - \dfrac{4 \pi}{3} \right)
\tan \left( - \dfrac{17 \pi}{6} \right)
\text{cosec } \left( - \dfrac{17 \pi}{6} \right)
\sec \left( - \dfrac{17 \pi}{6} \right)
\cot \left( - \dfrac{17 \pi}{6} \right)
\cot 3 \pi
Find the exact value of the following:
\dfrac{\left(\sin \dfrac{2 \pi}{3}\right) \left(\cos \dfrac{2 \pi}{3}\right) \left(\tan \dfrac{3 \pi}{4}\right)}{\tan \left( - \dfrac{\pi}{4} \right)}
\dfrac{\sin \dfrac{2 \pi}{3} + \cos \dfrac{5 \pi}{6} - \tan \dfrac{7 \pi}{4}}{\cos \dfrac{4 \pi}{3}}
Consider the equation y = \sin x.
Complete the table with values in exact form:
| x | 0 | \dfrac{\pi}{6} | \dfrac{\pi}{2} | \dfrac{5 \pi}{6} | \pi | \dfrac{7 \pi}{6} | \dfrac{3 \pi}{2} | \dfrac{11 \pi}{6} | 2 \pi |
|---|---|---|---|---|---|---|---|---|---|
| \sin x |
Sketch a graph for y = \sin x on the domain -2\pi \leq 0 \leq 2\pi.
State the value of \sin \left(-2 \pi\right).
State the sign of \sin \left( \dfrac{- \pi}{12} \right).
State the sign of \sin \dfrac{13 \pi}{12}.
Which quadrant of a unit circle does an angle with measure \dfrac{13 \pi}{12} lie in?
Consider the equation y = \cos x.
Complete the table with values in exact form:
| x | 0 | \dfrac{\pi}{3} | \dfrac{\pi}{2} | \dfrac{2 \pi}{3} | \pi | \dfrac{4 \pi}{3} | \dfrac{3 \pi}{2} | \dfrac{5 \pi}{3} | 2 \pi |
|---|---|---|---|---|---|---|---|---|---|
| \cos x |
Sketch a graph for y = \cos x on the domain -2\pi \leq 0 \leq 2\pi.
State the value of \cos \pi.
State the sign of \cos \left( \dfrac{- \pi}{4} \right).
State the sign of \cos \dfrac{11 \pi}{6}.
Which quadrant of a unit circle does an angle with measure \dfrac{11 \pi}{6} lie in?
Consider the equation y = \tan x.
Complete the table with values in exact form:
| x | 0 | \dfrac{\pi}{4} | \dfrac{\pi}{2} | \dfrac{3 \pi}{4} | \pi | \dfrac{5 \pi}{4} | \dfrac{3 \pi}{2} | \dfrac{7 \pi}{4} | 2 \pi |
|---|---|---|---|---|---|---|---|---|---|
| \tan x |
Sketch the graph of y = \tan x on the domain -2\pi \leq 0 \leq 2\pi.
Graph the line y = 1 on the same coordinate plane.
Hence, state the exact solutions to the equation \tan x = 1 over this domain.
State the value of \tan \left(-2 \pi\right).
State the sign of \tan \left( \dfrac{- \pi}{6} \right).
State the sign of \tan \dfrac{9 \pi}{5}.
Which quadrant of a unit circle does an angle with measure \dfrac{9 \pi}{5} lie in?
Consider the graph of y = - \tan x and the plotted points A, B, C, D and E shown:
At which point is the graph of \\ y = -\tan x equal to zero?
At which point will the corresponding graph of y = -\cot x be undefined? Explain your answer.
Consider the identity \sec x = \dfrac{1}{\cos x} and the following table of values:
| x | 0 | \dfrac{\pi}{4} | \dfrac{\pi}{2} | \dfrac{3 \pi}{4} | \pi | \dfrac{5 \pi}{4} | \dfrac{3 \pi}{2} | \dfrac{7 \pi}{4} | 2 \pi |
|---|---|---|---|---|---|---|---|---|---|
| \cos x | 1 | \dfrac{1}{\sqrt{2}} | 0 | - \dfrac{1}{\sqrt{2}} | - 1 | - \dfrac{1}{\sqrt{2}} | 0 | \dfrac{1}{\sqrt{2}} | 1 |
| \sec x |
Complete the table for the function y=\sec x.
State the range for y=\sec x.
Sketch the graph of y = \sec x on the domain \left[-2\pi, 2 \pi\right].
Consider the identity \text{cosec } x = \dfrac{1}{\sin x} and the following table of values:
| x | 0 | \dfrac{\pi}{4} | \dfrac{\pi}{2} | \dfrac{3 \pi}{4} | \pi | \dfrac{5 \pi}{4} | \dfrac{3 \pi}{2} | \dfrac{7 \pi}{4} | 2 \pi |
|---|---|---|---|---|---|---|---|---|---|
| \sin x | 0 | \dfrac{1}{\sqrt{2}} | 1 | \dfrac{1}{\sqrt{2}} | 0 | - \dfrac{1}{\sqrt{2}} | - 1 | - \dfrac{1}{\sqrt{2}} | 0 |
| \text{cosec } x |
State the values of x in the interval \left[0, 2 \pi\right] for which \text{cosec } x is not defined.
Complete the table for the function y=\text{cosec } x.
State the range for y=\text{cosec } x.
Sketch the graph of y = \text{cosec } x on the interval \left[-2\pi, 2 \pi\right].
Consider the identity \cot x = \dfrac{\cos x}{\sin x} and the following table of values:
| x | 0 | \dfrac{\pi}{4} | \dfrac{\pi}{2} | \dfrac{3 \pi}{4} | \pi | \dfrac{5 \pi}{4} | \dfrac{3 \pi}{2} | \dfrac{7 \pi}{4} | 2 \pi |
|---|---|---|---|---|---|---|---|---|---|
| \cos x | 1 | \dfrac{1}{\sqrt{2}} | 0 | - \dfrac{1}{\sqrt{2}} | - 1 | - \dfrac{1}{\sqrt{2}} | 0 | \dfrac{1}{\sqrt{2}} | 1 |
| \sin x | 0 | \dfrac{1}{\sqrt{2}} | 1 | \dfrac{1}{\sqrt{2}} | 0 | - \dfrac{1}{\sqrt{2}} | - 1 | - \dfrac{1}{\sqrt{2}} | 0 |
| \cot x |
State the values of x in the interval \left[0, 2 \pi\right] for which \cot x is not defined.
Complete the table for the function y = \cot x.
State the x-intercepts of the graph of y = \cot x in the interval \left[0, 2 \pi\right].
Sketch the graph of y = \cot x on the interval \left[-2\pi, 2 \pi\right].
Describe the graph of y = \cot x.
Consider the identity \sec x = \dfrac{1}{\cos x}.
Complete the table for x-values close to x = \dfrac{\pi}{2} \approx 1.57, where x is given in radians. Round each value to two decimal places.
| x | 1 | 1.5 | 1.56 | 1.57 | 1.58 | 1.6 | 2 |
|---|---|---|---|---|---|---|---|
| \sec x |
Describe happens to the value of \sec x as x approaches \dfrac{\pi}{2} from the left. Explain your answer.
Consider the identity \text{cosec } x = \dfrac{1}{\sin x}.
Complete the table for x-values close to x = \pi \approx 3.14, where x is given in radians. Round each value to two decimal places.
| x | 3 | 3.1 | 3.13 | 3.14 | 3.15 | 3.2 | 4 |
|---|---|---|---|---|---|---|---|
| \text{cosec } x |
Describe what happens to the value of \text{cosec } x as x approaches \pi from the left. Explain your answer.
Consider the table of values for \text{cosec } x over the domain \left(0, 2 \pi\right):
| x | \dfrac{\pi}{4} | \dfrac{\pi}{2} | \dfrac{3 \pi}{4} | \dfrac{5 \pi}{4} | \dfrac{3 \pi}{2} | \dfrac{7 \pi}{4} |
|---|---|---|---|---|---|---|
| \text{cosec } x | \sqrt{2} | 1 | \sqrt{2} | - \sqrt{2} | - 1 | - \sqrt{2} |
Given that the period of \text{cosec } x is 2 \pi, complete the table of values over the domain \left( - 2 \pi , 0\right):
| x | - \dfrac{7 \pi}{4} | - \dfrac{3 \pi}{2} | - \dfrac{5 \pi}{4} | - \dfrac{3 \pi}{4} | - \dfrac{\pi}{2} | - \dfrac{\pi}{4} |
|---|---|---|---|---|---|---|
| \text{cosec } x |
Sketch the graph of y = \text{cosec } x on the interval \left( - 4 \pi , 4 \pi\right).
Consider the table of values for \sec x in the interval \left[0, 2 \pi\right]:
| x | 0 | \dfrac{\pi}{4} | \dfrac{3 \pi}{4} | \pi | \dfrac{5 \pi}{4} | \dfrac{7 \pi}{4} | 2 \pi |
|---|---|---|---|---|---|---|---|
| \sec x | 1 | \sqrt{2} | - \sqrt{2} | - 1 | - \sqrt{2} | \sqrt{2} | 1 |
Given that the period of \sec x is 2 \pi, complete the table of values over the domain \left[ - 2 \pi , 0\right]:
| x | - 2 \pi | - \dfrac{7 \pi}{4} | - \dfrac{5 \pi}{4} | - \pi | - \dfrac{3 \pi}{4} | - \dfrac{\pi}{4} | 0 |
|---|---|---|---|---|---|---|---|
| \sec x | 1 |
Sketch the graph of y = \sec x on the interval \left[ - 4 \pi , 4 \pi\right].
Consider the graph of y = \text{cosec } x and the line y=\sqrt{2}:
When x = \dfrac{\pi}{4}, y = \sqrt{2}. Find the next positive x-value for which y = \sqrt{2}.
State the period of y = \text{cosec } x.
Find the smallest value of x greater than 2 \pi for which y = \sqrt{2}.
Find the first x-value less than 0 for which y = \sqrt{2}.
Consider the graph of y = \sec x:
When x = \dfrac{\pi}{3}, y = 2. Find the next positive x-value for which y = 2.
State the period of y = \sec x.
Find the smallest value of x greater than 2 \pi for which y = 2.
Find the first x-value less than 0 for which y = 2.
Consider the graph of y = \cot x and the line y=\sqrt{3}:
When x = \dfrac{\pi}{6}, y = \sqrt{3}. Find the next positive x-value for which y = \sqrt{3}.
State the period of y = \cot x.
Find the smallest value of x greater than 2 \pi for which y = \sqrt{3}.
Find the first x-value less than 0 for which y = \sqrt{3}.
Consider the graphs of \text{cosec } x and \sec x in the interval \left[-\dfrac{\pi}{2}, 2 \pi\right]:
State the interval where \text{cosec } x \gt 0 and \sec x \lt 0.