7.04 Reciprocal relationships
Write:
\text{cosec } \theta in terms of \sin \theta.
\text{sec } \theta in terms of \cos \theta.
\text{cot } \theta in terms of \tan \theta.
The point P on the unit circle has the coordinates \left(x, y\right):
Write the following in terms of x and/or y:
\text{cosec } \theta
\sec \theta
\cot \theta
Consider that \theta is an acute angle such that \tan \theta = x. Write an expression for the following in terms of x:
\sec \theta
\cot \theta
\cos \theta
\sin \theta
\text{cosec } \theta
Simplify:
\dfrac{\cos x}{\sin x}
\dfrac{\sin x }{\text{cosec } x }
\dfrac{\sec x}{\cos x}
Prove the following:
Find the exact value of \tan 70 \degree \cot 70 \degree.
Find \cot \theta given that \tan \theta = 11.
Consider the approximations: \cos 26 \degree = 0.8988 and \sin 26 \degree = 0.4384, find the value of the following, correct to four decimal places:
\text{cosec } 26 \degree
\sec 26 \degree
\cot26 \degree
Consider the graph of the unit circle shown:
Find the value of:
\text{cosec }30 \degree
\cos 60 \degree
\sec 60 \degree
\tan 30 \degree
\cot 30 \degree
\cot 90 \degree
\text{cosec } 90 \degree
For what values of \theta, where 0 \leq \theta \leq 90, are the following ratios undefined:
P\left(\dfrac{1}{2}, - \dfrac{\sqrt{3}}{2} \right) is a point on the unit circle that corresponds to the angle \theta, measured anticlockwise from the positive x-axis. Find the exact values of the following:
\sin \theta
\cos \theta
\tan \theta
\text{cosec } \theta
\sec \theta
\cot \theta
Find the exact values of the following:
\sec 0 \degree
\sec \left( - 30 \right) \degree
\sec 30 \degree
\cot 30 \degree
\sec 90 \degree
\text{cosec } 180 \degree
\cot 180 \degree
\cot 225 \degree
\text{cosec } 270 \degree
\text{cosec } 300 \degree
\sec 300 \degree
\sin 330 \degree
\cos 330 \degree
\tan 330 \degree
\text{cosec } 330 \degree
\sec 330 \degree
\cot 330 \degree
\cot 585 \degree
\sin 945 \degree
\tan 945 \degree
\text{cosec } 945 \degree
\sec 945 \degree
\cot 945 \degree
Find the exact value of the following trigonometric ratios. Write your answers with a rational denominators.
\text{cosec }60 \degree
\cot 45 \degree
State whether the following statements are true:
When \sin \theta = \dfrac{1}{3}, \text{cosec } \theta = 3.
When \sin \theta = \dfrac{1}{3}, \text{cosec } \theta = \dfrac{1}{3}.
When \sin \theta = 3, \text{cosec } \theta = \dfrac{1}{3}.
When \sin \theta = - \dfrac{1}{3}, \text{cosec } \theta = 3.
When \cos \theta \gt 0, \sec \theta \lt 0.
\cos \theta = \sec \theta
When \cos \theta \lt 0, \sec \theta \lt 0.
When \cos \theta \lt 0, \sec \theta \gt 0.
When \tan \theta = 3.5, \cot \theta = - 3.5.
If \sin \alpha = - \dfrac{4}{5} and \cos \alpha = \dfrac{3}{5}, find the exact values of:
\tan \alpha
\cot \alpha
\sec \alpha
\text{cosec } \alpha
If \sin \alpha = \dfrac{\sqrt{7}}{4} and \cos \alpha = \dfrac{3}{4}, find the exact values of:
\tan \alpha
\cot \alpha
\sec \alpha
\text{cosec } \alpha
If \cot \theta = 0.6 and \text{cosec } \theta \lt 0, find the exact values of:
\sin \theta
\text{cosec } \theta
\tan \theta
\sec \theta
If \sec \theta = - \dfrac{6}{5} and \sin \theta \gt 0, find the exact values of:
\tan \theta
\text{cosec }\theta
\cot \theta
Given the following, find the value of \sin \theta.
\text{cosec } \theta = \sqrt{2}
\text{cosec } \theta = - \dfrac{2 \sqrt{6}}{3}
Given the following, find the value of \text{cosec } \theta.
\sin \theta = \dfrac{2}{9}
\cot \theta = \dfrac{2}{\sqrt{3}} and - 90 \degree \leq \theta \leq 90 \degree
Given the following, find the value of \sec \theta.
\cos \theta = \dfrac{2}{9}
\text{cosec } \theta = \dfrac{3}{2}, \theta \gt 0 and obtuse
\cot \theta = \dfrac{9}{40}, \theta \gt 0 and reflex
Find \tan \theta given that \sec \theta = - \dfrac{17}{8} and 0 \degree \leq \theta \leq 180 \degree
Consider the angle 270 \degree:
State the coordinates of the point on the unit circle that corresponds to 270 \degree.
Find the value of \cos 270 \degree.
Find the value of \sin 270 \degree.
Determine whether the following are undefined:
\tan 270 \degree
\cot 270 \degree
\text{cosec }270 \degree
\sec 270 \degree
Consider the angle 450 \degree:
State the coordinates of the point on the unit circle that corresponds to 450 \degree.
Find the value of \cos 450 \degree.
Find the value of \sin 450 \degree.
Determine whether the following are undefined:
\cot 450 \degree
\tan 450 \degree
\text{cosec } 450 \degree
\sec 450 \degree
Consider the angle - 630 \degree:
State the coordinates of the point on the unit circle that corresponds to - 630 \degree.
Find the value of \cos \left( - 630 \degree\right).
Find the value of \sin \left( - 630 \degree\right).
Determine whether the following are undefined:
\sec \left( - 630 \degree\right)
\text{cosec } \left( - 630 \degree\right)
\tan \left( - 630 \degree\right)
\cot \left( - 630 \degree\right)
Show that \dfrac{\sec 30 \degree - \text{cosec } 300 \degree}{\cot 480 \degree + \tan \left( - 225 \degree \right)} = 2 - 2 \sqrt{3}.
Simplify:
\dfrac{\tan \left(90 \degree - x\right)}{\cot \left( - x \right)}
\cos \left(90 \degree - \theta\right) \text{cosec } \left(180 \degree + \theta\right)
Find the value of \theta for the following equations:
\tan \left(15 \degree - \theta\right) = \cot \left( 2 \theta + 60 \degree\right) where \theta \gt 0 and acute
\text{cosec } 32 \degree = \sec \left(\theta + 27 \degree\right) where 0 \degree \leq \left(\theta + 27 \degree\right) \leq 90 \degree
\cot \left( 2 \theta + 5 \degree\right) = \tan \left( 3 \theta - 15 \degree\right) where \theta \gt 0and acute
\text{cosec } \theta = \sec 24 \degree where 0 \degree \leq \theta \leq 90 \degree
\tan 24 \degree = \cot \theta where 0 \degree \leq \theta \leq 90 \degree
\tan \left( 3 \theta - 5 \degree\right) = \dfrac{1}{\cot \left( 7 \theta - 53 \degree\right)}
State whether the following values are positive or negative:
\sec 324 \degree
\text{cosec } 107 \degree
\tan 162 \degree
Consider the identity \sec x = \dfrac{1}{\cos x}:
Complete the table of values, writing '-' if the value is undefined:
| x | 0\degree | 45\degree | 90\degree | 135\degree | 180\degree | 225\degree | 270\degree | 315\degree | 360\degree |
|---|---|---|---|---|---|---|---|---|---|
| \cos x | |||||||||
| \sec x |
State the minimum positive value of \sec x.
State the maximum negative value of \sec x.
Sketch the graph of y = \sec x and y = \cos x for 0 \leq x \leq 360 \degree on the same number plane.
Consider the identity \text{cosec } x = \dfrac{1}{\sin x}.
Complete the table of values, writing '-' if the value is undefined:
| x | 0\degree | 45\degree | 90\degree | 135\degree | 180\degree | 225\degree | 270\degree | 315\degree | 360\degree |
|---|---|---|---|---|---|---|---|---|---|
| \sin x | |||||||||
| \text{cosec } x |
State the minimum positive value of \text{cosec } x.
State the maximum negative value of \text{cosec } x.
Sketch the graph of y = \text{cosec } x and y = \sin x for 0 \leq x \leq 360 \degree on the same number plane.
Consider the identity \cot x = \dfrac{\cos x}{\sin x}.
Complete the table of values, writing '-' if the value is undefined:
| x | 0\degree | 45\degree | 90\degree | 135\degree | 180\degree | 225\degree | 270\degree | 315\degree | 360\degree |
|---|---|---|---|---|---|---|---|---|---|
| \cot x |
Find the x-intercepts of the graph of y = \cot x in the interval \left[0 \degree, 360 \degree\right].
Sketch the graph of y = \cot x for 0 \leq x \leq 360 \degree.
Consider the identity \sec x = \dfrac{1}{\cos x}.
Complete the table of values, rounding each value to two decimal places:
| x | 57.30\degree | 85.94\degree | 89.38\degree | 89.95\degree | 90.53\degree | 91.67\degree | 114.59\degree |
|---|---|---|---|---|---|---|---|
| \sec x |
What happens to the value of \sec x when x approaches 90 \degree from the left? Explain your answer.
Consider the identity \text{cosec } x = \dfrac{1}{\sin x}.
Complete the table of values, rounding each value to two decimal places:
| x | 171.89\degree | 177.62\degree | 179.34\degree | 179.91\degree | 180.48\degree | 183.35\degree | 229.18\degree |
|---|---|---|---|---|---|---|---|
| \text{cosec }x |
What happens to the value of \text{cosec } x when x approaches 180 \degree from the left? Explain your answer.
Consider the function y = \sec x:
If x = 45 \degree, y = \sqrt{2}. Find the next positive x-value for which y = \sqrt{2}.
Find the period of the graph.
Find the smallest value of x greater than 360 \degree for which y = \sqrt{2}.
Find the first x-value less than 0 \degree for which y = \sqrt{2}.
Consider the function y = \text{cosec } x:
If x = 30 \degree, y = 2. Find the the next positive x-value for which y = 2.
Find the period of the graph.
Find the smallest value of x greater than 360 \degree for which y = 2.
Find the first x-value less than 0 \degree for which y = 2.
Consider the function y = \cot x:
If x = 30 \degree, y = \dfrac{1}{\sqrt{3}}.Find the next positive x-value for which y = \dfrac{1}{\sqrt{3}}.
Find the period of the graph.
Find the smallest value of x greater than 360 \degree for which y = \dfrac{1}{\sqrt{3}}.
Find the first x-value less than 0 \degree for which y = \dfrac{1}{\sqrt{3}}.
Consider the functions y=\text{cosec } x and y=\sec x shown below. State the domain for which \\ \text{cosec } x \lt 0, \, \sec x \gt 0 and 0 \leq x \leq 360.