Trigonometric Functions

Lesson

Having worked with the primary trigonometric functions $\sin x$`s``i``n``x`, $\cos x$`c``o``s``x` and $\tan x$`t``a``n``x`, we can now look at the closely related reciprocal trigonometric functions.

Recall that the reciprocal of the number $a$`a` is just the number $\frac{1}{a}$1`a`, provided $a$`a` is not equal to zero. Here is a summary of the reciprocal trigonometric ratios, and how they relate to the sides of a right triangle.

Reciprocal trigonometric ratios

Cosecant: $\csc(\theta)=\frac{1}{\sin(\theta)}=\frac{\text{Hypotenuse}}{\text{Opposite}}$ Secant: $\sec(\theta)=\frac{1}{\cos(\theta)}=\frac{\text{Hypotenuse}}{\text{Adjacent}}$ Cotangent: $\cot(\theta)=\frac{1}{\tan(\theta)}=\frac{\text{Adjacent}}{\text{Opposite}}$ |

Find the exact value of $f\left(x\right)=\sec x$`f`(`x`)=`s``e``c``x` when $x=\frac{\pi}{3}$`x`=π3.

**Think**: We want to find the value of $\sec\frac{\pi}{3}$`s``e``c`π3, and express the result as an exact value (a rational number or multiple of a surd). One approach is to first find $\cos\frac{\pi}{3}$`c``o``s`π3 and then take the reciprocal of that value. We can use the exact value triangles below to help us.

**Do**:

$\sec\frac{\pi}{3}$secπ3 |
$=$= | $\frac{1}{\cos\frac{\pi}{3}}$1cosπ3 |

$=$= | $\frac{1}{\frac{1}{2}}$112 | |

$=$= | $2$2 |

**Reflect**: We used the fact that $\cos\frac{\pi}{3}=\frac{1}{2}$`c``o``s`π3=12 to work out that $\sec\frac{\pi}{3}=2$`s``e``c`π3=2. In this example we worked with the exact values of the cosine function, but we can also use the exact value triangles directly to find the reciprocal function values. Since the secant of an angle is the ratio of the Hypotenuse to the Adjacent side, from the image above we have that $\sec\frac{\pi}{3}=\frac{2}{1}=2$`s``e``c`π3=21=2, as expected.

The function $f\left(x\right)$`f`(`x`) is defined as $f\left(x\right)=\csc x$`f`(`x`)=`c``s``c``x`.

Find the exact value of $f\left(\frac{\pi}{3}\right)$`f`(π3).

The function $f\left(x\right)$`f`(`x`) is defined as $f\left(x\right)=\sec x$`f`(`x`)=`s``e``c``x`.

Find the value of $f\left(\frac{99\pi}{34}\right)$`f`(99π34), rounded to three decimal places.

The function $f\left(x\right)$`f`(`x`) is defined as $f\left(x\right)=2\csc x-15$`f`(`x`)=2`c``s``c``x`−15.

Find the exact value of $f\left(\frac{\pi}{3}\right)$`f`(π3).

Form and use trigonometric, polynomial, and other non-linear equations

Apply trigonometric methods in solving problems