7.01 Evaluating trigonometric functions with a calculator

Lesson

Although many trigonometric values can be determined using the unit circle, the special right triangles, or the trigonometric identities, many other angles will not produce a "nice" exact value for the trig functions. For these, we use a calculator and round as required. It is important to note whether the angle value is in degrees or radians and set your calculator accordingly!

Evaluate trigonometric function values in degrees

If the angle value is given in degrees, the first step is to set your calculator to degree mode. Then, evaluating the function is simply a matter of choosing sine, cosine, or tangent.

Evaluate trigonometric function values in radians

If the angle value is given in radians, follow the same steps as above - but be sure to set your calculator to radian mode ahead of attempting the problem.

Evaluate trigonometric reciprocal function values

Having worked with the primary trigonometric functions $\sin x$sinx, $\cos x$cosx and $\tan x$tanx, 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}$1a, provided $a$a is not equal to zero. Here is a quick review of the reciprocal trigonometric ratios, and how they relate to the sides of a right triangle.

Reciprocal trigonometric ratios

Cosecant: $\csc(\theta)=\frac{1}{\sin(\theta)}=\frac{\text{Hypotenuse}}{\text{Opposite}}$csc(θ)=1sin(θ)=HypotenuseOpposite

Secant: $\sec(\theta)=\frac{1}{\cos(\theta)}=\frac{\text{Hypotenuse}}{\text{Adjacent}}$sec(θ)=1cos(θ)=HypotenuseAdjacent

Cotangent: $\cot(\theta)=\frac{1}{\tan(\theta)}=\frac{\text{Adjacent}}{\text{Opposite}}$cot(θ)=1tan(θ)=AdjacentOpposite

We'll use these relationships to find the reciprocal trig function values using a calculator.

Worked example

Question 1

Find the value of $\sec\frac{\pi}{8}$secπ8, rounded to three decimal places.

Think: We want to find the value of $\sec\frac{\pi}{8}$secπ8, and express the result as a decimal. One approach is to first rewrite $\sec\frac{\pi}{8}$secπ8 as its reciprocal. That is, rewrite the secant function as a cosine.

Do: Rewrite $\sec\frac{\pi}{8}=\frac{1}{\cos\frac{\pi}{8}}$secπ8=1cosπ8

$\frac{\pi}{8}$π8 is a radian value, so set your calculator to radian mode. Then use the calculator to determine $\frac{1}{\cos\frac{\pi}{8}}=1.08239220029$1cosπ8=1.08239220029... Rounding to three decimals gives us $\sec\frac{\pi}{8}=1.082$secπ8=1.082.

Reflect: Does this value for secant make sense? Would we have obtained the same answer by taking the reciprocal of the angle first?

Practice problems

Question 2

Evaluate the following expression, writing your answer correct to 2 decimal places.

$\sin69^\circ$sin69°

QUESTION 3

Evaluate the following expression using your calculator. Write your answer correct to two decimal places.

$\frac{\sin40^\circ}{\cos69^\circ+\tan68^\circ}$sin40°cos69°+tan68°

QUESTION 4

The function $f\left(x\right)$f(x) is defined as $f\left(x\right)=\sec x$f(x)=secx.

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