Year 9 Pythagoras and trigonometry: revision mind map

1Pythagoras' theorem

In a right-angled triangle: $c^2 = a^2 + b^2$. $c$ is the hypotenuse, the longest side.

Sides 6 and 8: $c^2 = 36 + 64 = 100$, so $c = 10$

2Finding a shorter side

To find a shorter side, subtract: $a^2 = c^2 - b^2$.

Hypotenuse 13, side 5: $a^2 = 169 - 25 = 144$, so $a = 12$

Watch out: a shorter side can never be longer than the hypotenuse.

3Pythagoras in 2D problems

Draw it, mark the right angle, then find the triangle.

A 5 m ladder reaches 4 m up a wall: the foot is $\sqrt{25 - 16} = 3$ m out

Diagonal of a 9 by 12 rectangle: $\sqrt{81 + 144} = 15$

4Pythagoras in 3D

Find a right triangle on one face first, then use that length in a second triangle.

Box 3 by 4 by 12: base diagonal $\sqrt{9 + 16} = 5$

Space diagonal $\sqrt{5^2 + 12^2} = 13$

Shortcut: $d = \sqrt{a^2 + b^2 + c^2}$

5Trigonometric ratios

SOH CAH TOA. Label the sides from the angle you're using.

$\sin\theta = \dfrac{O}{H}$   $\cos\theta = \dfrac{A}{H}$   $\tan\theta = \dfrac{O}{A}$

Watch out: set your calculator to degrees.

6Finding unknown sides

Pick the ratio that uses the side you know and the side you want.

Opposite 35°, hypotenuse 10: $x = 10\sin 35^\circ \approx 5.74$

Adjacent 8, angle 40°, find opposite: $x = 8\tan 40^\circ \approx 6.71$

7Solving for the denominator

When $x$ is on the bottom, swap it with the trig value.

$\cos 30^\circ = \dfrac{12}{x} \Rightarrow x = \dfrac{12}{\cos 30^\circ} \approx 13.86$

Watch out: check the answer makes sense: the hypotenuse must be the longest side.

8Finding unknown angles

Use the inverse buttons: $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$.

Opposite 5, hypotenuse 10: $\sin\theta = 0.5$, so $\theta = 30^\circ$

Opposite 7, adjacent 4: $\theta = \tan^{-1}(1.75) \approx 60.3^\circ$

9Applying trigonometry

Angles of elevation (looking up) and depression (looking down) are measured from the horizontal.

20 m from a tree, the angle of elevation to the top is 32°

Height $= 20\tan 32^\circ \approx 12.5$ m

Watch out: draw the horizontal line first, then the angle.

10Bearings

True bearings are measured clockwise from north, with three digits.

North-east is 045°T, south-east 135°T, west 270°T

Walk 10 km on 060°T: east $10\sin 60^\circ \approx 8.66$ km, north $10\cos 60^\circ = 5$ km