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Special Angles

5.5 Calculator skills (EMA3R)

In this section we will look at using a calculator to determine the values of the trigonometric ratios for any angle. For example we might want to know what the value of \(\sin 55^{\circ}\) is or what the value of \(\sec 34^{\circ}\) is.

When doing calculations involving the reciprocal ratios you need to convert the reciprocal ratio to one of the standard trigonometric ratios: \(\sin\), \(\cos\) and \(\tan\) as this is the only way to calculate these ratios on your calculator.

Most scientific calculators are quite similar but these steps might differ depending on the calculator you use. Make sure your calculator is in “degrees” mode.

Note that \(\sin^{2}\theta = (\sin \theta)^{2}\). This also applies for the other trigonometric ratios.

Worked example 2: Using your calculator

Use your calculator to calculate the following (correct to \(\text{2}\) decimal places):

  1. \(\cos 48°\)

  2. \(2\sin 35°\)

  3. \({\tan}^{2}81°\)

  4. \(3{\sin}^{2}72°\)

  5. \(\frac{1}{4}\cos 27°\)

  6. \(\frac{5}{6}\tan 34°\)

  7. \(\sec 34°\)

  8. \(\cot 49°\)

The following shows the keys to press on a Casio calculator. Other calculators work in a similar way. On a Casio calculator ( is automatically added after pressing \(\sin\), \(\cos\) and \(\tan\) so you just need to press \(\boxed{)}\) after typing in the angle to close the brackets.

  1. Press \(\boxed{\text{cos}} \enspace \boxed{48} \enspace \boxed{)} \enspace \boxed{=}\) \(\text{0,66913...}\) \(\approx\) \(\text{0,67}\)

  2. Press \(\boxed{2} \enspace \boxed{\text{sin}} \enspace \boxed{35} \enspace \boxed{)} \enspace \boxed{=} \enspace\) \(\text{1,147152...}\) \(\approx\) \(\text{1,15}\)

  3. Press \(\boxed{(} \enspace \boxed{\text{tan}} \enspace \boxed{81} \enspace \boxed{)} \enspace \boxed{)} \enspace \boxed{x^{2}} \enspace \boxed{=} \enspace\) \(\text{39,86345...}\) \(\approx\) \(\text{39,86}\)

    OR

    Press \(\boxed{\text{tan}} \enspace \boxed{81} \enspace \boxed{)} \enspace \boxed{=} \enspace \boxed{\text{ANS}} \enspace \boxed{x^{2}} \enspace \boxed{\text{ANS}} \enspace \boxed{=} \enspace\) \(\text{39,86345...}\) \(\approx\) \(\text{39,86}\)

  4. Press \(\boxed{3} \enspace \boxed{(} \enspace \boxed{\text{sin}} \enspace \boxed{72} \enspace \boxed{)} \enspace \boxed{)} \enspace \boxed{x^{2}} \enspace \boxed{=} \enspace\) \(\text{2,71352...}\) \(\approx\) \(\text{2,71}\)

    OR

    Press \(\boxed{\text{sin}} \enspace \boxed{72} \enspace \boxed{)} \enspace \boxed{ =} \enspace \boxed{\text{ANS}} \enspace \boxed{x^{2}} \enspace \boxed{=} \enspace \boxed{\text{ANS}} \enspace \boxed{\times} \enspace \boxed{3}\)

  5. Press \(\boxed{(} \enspace \boxed{1} \enspace \boxed{\div} \enspace \boxed{4} \enspace \boxed{)} \enspace \boxed{\text{cos}} \enspace \boxed{27} \enspace \boxed{)} \enspace \boxed{=} \enspace\)\(\text{0,22275...}\) \(\approx\) \(\text{0,22}\)

    OR

    Press \(\boxed{\text{cos}} \enspace \boxed{27} \enspace \boxed{)} \enspace \boxed{=} \enspace \boxed{\text{ANS}} \enspace \boxed{\div} \enspace \boxed{4} \enspace \boxed{=} \enspace\) \(\text{0,22275...}\) \(\approx\) \(\text{0,22}\)

  6. Press \(\boxed{(} \enspace \boxed{5} \enspace \boxed{\div} \enspace \boxed{6} \enspace \boxed{)} \enspace \boxed{\text{tan}} \enspace \boxed{34} \enspace \boxed{)} \enspace \boxed{=} \enspace\) \(\text{0,56209...}\) \(\approx\) \(\text{0,56}\)

    OR

    Press \(\boxed{\text{tan}} \enspace \boxed{34} \enspace \boxed{=} \enspace \boxed{\text{ANS}} \enspace \boxed{\times} \enspace \boxed{5} \enspace \boxed{\div} \enspace \boxed{6} \enspace \boxed{=} \enspace\) \(\text{0,56}\) \(\approx\) \(\text{0,56}\)

  7. First write \(\sec\) in terms of \(\cos\): \(\sec 34° = \dfrac{1}{\cos 34°}\) (since there is no “\(\sec\)” button on your calculator).

    Press \(\boxed{1} \enspace \boxed{\div} \enspace \boxed{(} \enspace \boxed{\text{cos}} \enspace \boxed{34} \enspace \boxed{)} \enspace \boxed{)} \enspace \boxed{=} \enspace\) \(\text{1,206217...}\) \(\approx\) \(\text{1,21}\)

  8. First write \(\cot\) in terms of \(\tan\): \(\cot 49° = \dfrac{1}{\tan 49°}\) (since there is no “\(\cot\)” button on your calculator).

    Press \(\boxed{1} \enspace \boxed{\div} \enspace \boxed{(} \enspace \boxed{\text{tan}} \enspace \boxed{49} \enspace \boxed{)} \enspace \boxed{)} \enspace \boxed{=} \enspace\) \(\text{0,869286...}\) \(\approx\) \(\text{0,87}\)

Worked example 3: Calculator work using substitution

If \(x = 25°\) and \(y = 65°\), use your calculator to determine whether the following statement is true or false:

\[\sin^{2}x + \cos^{2}\left(90°-y\right) = 1\]

Calculate the left hand side of the equation

Press \(\boxed{(} \enspace \boxed{\text{sin}} \enspace \boxed{25} \enspace \boxed{)} \enspace \boxed{)} \enspace \boxed{x^{2}} \enspace \boxed{+} \enspace \boxed{(} \enspace \boxed{\text{cos}} \enspace \boxed{90} \enspace \boxed{-} \enspace \boxed{65} \enspace \boxed{)} \enspace \boxed{)} \enspace \boxed{x^{2}} \enspace \boxed{=} \enspace \text{1}\)

Write the final answer

LHS = RHS therefore the statement is true.

Exercise 5.2

Use your calculator to determine the value of the following (correct to \(\text{2}\) decimal places):

\(\tan 65°\)

\begin{align*} \tan 65° & = \text{2,1445069...} \\ & \approx \text{2,14} \end{align*}

\(\sin 38°\)

\begin{align*} \sin 38° & = \text{0,615661...} \\ & \approx \text{0,62} \end{align*}

\(\cos 74°\)

\begin{align*} \cos 74° & = \text{0,275637...} \\ & \approx \text{0,28} \end{align*}

\(\sin 12°\)

\begin{align*} \sin 12° & = \text{0,20791...} \\ & \approx \text{0,21} \end{align*}

\(\cos 26°\)

\begin{align*} \cos 26° & = \text{0,898794...} \\ & \approx \text{0,90} \end{align*}

\(\tan 49°\)

\begin{align*} \tan 49° & = \text{1,150368...} \\ & \approx \text{1,15} \end{align*}

\(\sin 305°\)

\begin{align*} \sin 305° & = -\text{0,81915...} \\ & \approx -\text{0,82} \end{align*}

\(\tan 124°\)

\begin{align*} \tan 124° & = -\text{1,482560...} \\ & \approx -\text{1,48} \end{align*}

\(\sec 65°\)

\begin{align*} \sec{65°} & = \frac{1}{\cos{65°}} \\ & = \text{2,36620...} \\ & \approx \text{2,37} \end{align*}

\(\sec 10°\)

\begin{align*} \sec{10°} & = \frac{1}{\cos{10°}} \\ & = \text{1,01542...} \\ & \approx \text{1,02} \end{align*}

\(\sec{48°}\)

\begin{align*} \sec{48°} & = \frac{1}{\cos{48°}} \\ & = \text{1,49447...} \\ & \approx \text{1,49} \end{align*}

\(\cot{32°}\)

\begin{align*} \cot{32°} & = \frac{1}{\tan{32°}} \\ & = \text{1,6003334...} \\ & \approx \text{1,60} \end{align*}

\(\text{cosec }140°\)

\begin{align*} \text{cosec } 140° & = \frac{1}{\sin{140°}} \\ & = \text{1,555724...} \\ & \approx \text{1,56} \end{align*}

\(\text{cosec }237°\)

\begin{align*} \text{cosec } 237° & = \frac{1}{\sin{237°}} \\ & = -\text{1,192363...} \\ & \approx -\text{1,19} \end{align*}

\(\sec 231°\)

\begin{align*} \sec 231° & = \frac{1}{\cos{231°}} \\ & = -\text{1,589016...} \\ & \approx -\text{1,59} \end{align*}

\(\text{cosec }226°\)

\begin{align*} \text{cosec } 226° & = \frac{1}{\sin{226°}} \\ & = -\text{1,390164...} \\ & \approx -\text{1,39} \end{align*}

\(\dfrac{1}{4}\cos 20°\)

\begin{align*} \frac{1}{4}\cos 20° & = \frac{1}{4}(\text{0,939692...}) \\ & = \text{0,234923...} \\ & \approx \text{0,23} \end{align*}

\(3\tan 40°\)

\begin{align*} 3\tan 40° & = 3(\text{0,83909963...}) \\ & = \text{2,517298894...} \\ & \approx \text{2,52} \end{align*}

\(\dfrac{2}{3}\sin 90°\)

\begin{align*} \frac{2}{3}\sin 90° & = \frac{2}{3}(\text{1}) \\ & = \text{0,66666...} \\ & \approx \text{0,67} \end{align*}

\(\dfrac{5}{\cos \text{4,3}°}\)

\begin{align*} \frac{5}{\cos \text{4,3}°} & = \frac{5}{\text{0,9971...}} \\ & \approx \text{5,01} \end{align*}

\(\sqrt{\sin 55°}\)

\begin{align*} \sqrt{\sin 55°} & = \sqrt{\text{0,81915...}} \\ & \approx \text{0,91} \end{align*}

\(\dfrac{\sin 90°}{\cos 90°}\)

\begin{align*} \frac{\sin 90°}{\cos 90°} & = \frac{1}{0} \\ & \text{undefined} \end{align*}

\(\tan 35° + \cot 35°\)

\begin{align*} \tan 35° + \cot 35° & = \text{0,7002...} + \frac{1}{\tan{35°}} \\ & = \text{0,7002...} + \text{1,4281...} \\ & \approx \text{2,13} \end{align*}

\(\dfrac{2 + \cos 310°}{2 + \sin 87°}\)

\begin{align*} \frac{2 + \cos 310°}{2 + \sin 87°} & = \frac{\text{2,64278...}}{\text{2,99862...}} \\ & \approx \text{0,88} \end{align*}

\(\sqrt{4 \sec 99°}\)

\begin{align*} \sqrt{4 \sec 99°} & = \sqrt{\frac{4}{\cos 99°}} \\ & = \sqrt{-\text{25,5698...}} \\ & \text{non-real} \end{align*}

\(\sqrt{\dfrac{\cot 103° + \sin 1090°}{\sec 10° + 5}}\)

\begin{align*} \sqrt{\frac{\cot 85° + \sin 1090°}{\sec 10° + 5}} & = \sqrt{\frac{\frac{1}{\tan 85°} + \sin 1090°}{\frac{1}{\cos 10°} + 5}} \\ & = \sqrt{\frac{\text{0,2611...}}{\text{6,015...}}} \\ & = \sqrt{\text{0,043411...}} \\ & \approx \text{0,21} \end{align*}

If \(x = 39°\) and \(y=21°\), use a calculator to determine whether the following statements are true or false:

\(\cos x + 2\cos x = 3\cos x\)

LHS:

\begin{align*} \cos x + 2\cos x & = \cos{39°} + 2\cos{39°} \\ & = \text{0,7771...} + \text{1,55429...} \\ & = \text{2,3314...} \\ & \approx \text{2,33} \end{align*}

RHS:

\begin{align*} 3\cos x & = 3 \cos{39°} \\ & = \text{2,3314...} \\ & \approx \text{2,33} \end{align*}

Therefore the statement is true.

\(\cos 2y = \cos y + \cos y\)

LHS:

\begin{align*} \cos 2y & = \cos{2(21°}) \\ & = \text{0,7431...} \\ & \approx \text{0,74} \end{align*}

RHS:

\begin{align*} \cos y + \cos y & = \cos{21°} + \cos{21°} \\ & = \text{0,93358...} + \text{0,93358...} \\ & = \text{1,86716...} \\ & \approx \text{1,86} \end{align*}

Therefore the statement is false.

\(\tan x = \dfrac{\sin x}{\cos x}\)

LHS:

\begin{align*} \tan x & = \tan{39°} \\ & = \text{0,809784...} \\ & \approx \text{0,81} \end{align*}

RHS:

\begin{align*} \frac{\sin x}{\cos x} & = \frac{\sin{39°}}{\cos{39°}} \\ & = \frac{\text{0,62932...}}{\text{0,777145...}} \\ & = \text{0,80978...} \\ & \approx \text{0,81} \end{align*}

Therefore the statement is true.

\(\cos(x + y) = \cos x + \cos y\)

LHS:

\begin{align*} \cos (x + y) & = \cos{39° + 21°} \\ & \approx \text{0,5} \end{align*}

RHS:

\begin{align*} \cos x + \cos y & = \cos{39°} + \cos{21°} \\ & = \text{0,777145...} + \text{0,933358...} \\ & = \text{1,71072...} \\ & \approx \text{1,71} \end{align*}

Therefore the statement is false.

Solve for \(x\) in \(5^{\tan x} = 125\).

To solve this problem we need to recall from exponents that if \(a^{x} = a^{y}\) then \(x = y\). Then we note that \(125 = 5^{3}\). Now we can solve the problem:

\begin{align*} 5^{\tan x} & = 5^{3} \\ \therefore \tan x & = 3 \\ x & = \text{71,56505...} \\ & \approx \text{71,57} \end{align*}