Year 9AC9M9M03

Calculate an Identity Without a Calculator

Solve spatial problems, applying angle properties, scale, similarity, Pythagoras' theorem and trigonometry in right-angled triangles.

Use Pythagoras to find the missing side, then read off any ratio you need.

Worked example

e.g.

\text{determine } \cos\theta \text{ and } \tan\theta \text{ without a calculator if } 5\sin\theta - 3 = 0

Get the trig ratio on its own $\sin\theta = \tfrac{3}{5}$
Draw a right angled triangle and fill in what you know $opp = 3,\ hyp = 5$
Using Pythagoras, calculate the unknown side $adj = \sqrt{5^{2} - 3^{2}} = 4$
Read the other ratios off the triangle $\therefore \cos\theta = \tfrac{4}{5} \text{ and } \tan\theta = \tfrac{3}{4}$

Determine the value of:

\cos\theta \text{ if } \sin\theta = \tfrac{1}{2}

Answer

\tfrac{\sqrt{3}}{2}

\tan\theta \text{ if } \cos\theta = \tfrac{\sqrt{3}}{2}

Answer

\tfrac{\sqrt{3}}{3}

\cos\theta \text{ if } \sin\theta = \tfrac{\sqrt{2}}{2}

Answer

\tfrac{\sqrt{2}}{2}

\tan\theta \text{ if } \cos\theta = \tfrac{3}{5}

Answer

\tfrac{4}{3}

\cos\theta \text{ if } \tan\theta = \tfrac{1}{2}

Answer

\tfrac{2\sqrt{5}}{5}

\cos\theta \text{ if } 5\sin\theta = 3

Answer

\tfrac{4}{5}

\tan\theta \text{ if } 5\cos\theta = 4

Answer

\tfrac{3}{4}

\cos\theta \text{ if } 2\tan\theta = \sqrt{3}

Answer

\tfrac{2\sqrt{7}}{7}

\sin\theta \text{ if } 3\tan\theta = 2

Answer

\tfrac{2\sqrt{13}}{13}

\cos\theta \text{ if } 9\sin\theta = 5

Answer

\tfrac{2\sqrt{14}}{9}