Pythagoras & Trigonometry Flashcards

All 22 cards in this deck

What is Pythagoras' theorem?

$a^2 + b^2 = c^2$
$c$ is the hypotenuse (longest side, opposite the right angle); $a$ and $b$ are the two shorter sides.

What are the steps to find the hypotenuse of a right-angled triangle?

e.g. shorter sides 6 cm and 8 cm

1. Square both shorter sides: $6^2 = 36$, $8^2 = 64$
2. Add them: $36 + 64 = 100$
3. Square root: $c = \sqrt{100} = 10$ cm

True or false? If you know the hypotenuse and one shorter side, you find the other shorter side by subtracting the squares.

True.
e.g. hypotenuse 8.5, shorter side 4: $8.5^2 - 4^2 = 56.25$, so the side is $7.5$.

How do you test whether a triangle is right-angled using Pythagoras' theorem?

e.g. sides 9 cm, 12 cm, 15 cm

Square the two shorter sides, add, and check it equals the square of the longest side.
$9^2 + 12^2 = 225 = 15^2$, so it is right-angled.

What are the steps to find the distance between two points on a coordinate grid?

e.g. $(-1, 1)$ and $(3, 4)$

1. Find the difference in $x$ and in $y$: $4$ and $3$
2. Square and add: $16 + 9 = 25$
3. Square root: distance $= \sqrt{25} = 5$

What are the steps to find the perpendicular height of an equilateral triangle, leaving the answer in surd form?

e.g. side 8 cm

1. Split it into two right-angled triangles: base $= 4$, hypotenuse $= 8$
2. Pythagoras: $h^2 = 8^2 - 4^2 = 48$
3. $h = \sqrt{48} = 4\sqrt{3}$ cm

What do the three trigonometric ratios SOH CAH TOA stand for?

$\sin\theta = \dfrac{\text{opp}}{\text{hyp}}$, $\cos\theta = \dfrac{\text{adj}}{\text{hyp}}$, $\tan\theta = \dfrac{\text{opp}}{\text{adj}}$

In a right-angled triangle, how do you identify the opposite, adjacent and hypotenuse sides?

Hypotenuse: opposite the right angle.
Opposite: facing the marked angle. Adjacent: the remaining side, next to the marked angle.

What are the steps to find a missing side using trigonometry?

e.g. hypotenuse 15 cm, angle $23^\circ$, find the opposite side $AB$

1. Label the sides: opposite $= AB$, hypotenuse $= 15$, so use sin
2. Write the equation: $\sin 23^\circ = \dfrac{AB}{15}$
3. Rearrange: $AB = 15 \times \sin 23^\circ = 5.86$ cm (3 s.f.)

What are the steps to find a missing angle in a right-angled triangle?

e.g. adjacent 15 cm, hypotenuse 18 cm, find angle $x$

1. Label the sides: adjacent and hypotenuse, so use cos
2. Write the equation: $\cos x = \dfrac{15}{18}$
3. Use the inverse: $x = \cos^{-1}\left(\dfrac{15}{18}\right) = 33.6^\circ$ (1 d.p.)

True or false? The value of $\sin\theta$ for an acute angle can be greater than 1.

False.
Sine and cosine of an acute angle are always between 0 and 1 (the opposite/adjacent side is shorter than the hypotenuse); only tan can exceed 1.

What is an angle of elevation?

The angle measured upwards from the horizontal to the line of sight of an object above.

What is an angle of depression?

The angle measured downwards from the horizontal to the line of sight of an object below.

True or false? The angle of depression from the top of a cliff to a boat equals the angle of elevation from the boat to the top of the cliff.

True.
The two horizontals are parallel, so the angles are alternate angles.

What are the steps to find a height from an angle of elevation?

e.g. standing 20 m from the foot of a tower, the angle of elevation of the top is $35^\circ$

1. Sketch the right-angled triangle: horizontal 20 m, angle $35^\circ$ at the observer, height $h$ opposite
2. Choose the ratio: $\tan 35^\circ = \dfrac{h}{20}$
3. Solve: $h = 20\tan 35^\circ = 14.0$ m (3 s.f.)

What are the steps to find an angle of elevation?

e.g. a tower 15 m high seen from a point 20 m from its foot

1. Sketch the triangle: opposite $= 15$, adjacent $= 20$, so use tan
2. Write: $\tan\theta = \dfrac{15}{20}$
3. Inverse: $\theta = \tan^{-1}(0.75) = 36.9^\circ$ (1 d.p.)

What are the exact values of $\sin 30^\circ$, $\sin 45^\circ$ and $\sin 60^\circ$?

$\sin 30^\circ = \dfrac{1}{2}$, $\sin 45^\circ = \dfrac{\sqrt{2}}{2}$, $\sin 60^\circ = \dfrac{\sqrt{3}}{2}$

What are the exact values of $\cos 30^\circ$, $\cos 45^\circ$ and $\cos 60^\circ$?

$\cos 30^\circ = \dfrac{\sqrt{3}}{2}$, $\cos 45^\circ = \dfrac{\sqrt{2}}{2}$, $\cos 60^\circ = \dfrac{1}{2}$

What are the exact values of $\tan 0^\circ$, $\tan 30^\circ$, $\tan 45^\circ$ and $\tan 60^\circ$?

$\tan 0^\circ = 0$, $\tan 30^\circ = \dfrac{1}{\sqrt{3}}$, $\tan 45^\circ = 1$, $\tan 60^\circ = \sqrt{3}$

What are the exact values of $\sin 0^\circ$, $\sin 90^\circ$, $\cos 0^\circ$ and $\cos 90^\circ$?

$\sin 0^\circ = 0$, $\sin 90^\circ = 1$, $\cos 0^\circ = 1$, $\cos 90^\circ = 0$

True or false? $\sin 30^\circ = \dfrac{\sqrt{3}}{2}$

False.
$\sin 30^\circ = \dfrac{1}{2}$; $\dfrac{\sqrt{3}}{2}$ is $\sin 60^\circ$.

What are the steps to find a side exactly using an exact trig value (no calculator)?

e.g. hypotenuse 6 cm, angle $30^\circ$, find the opposite side

1. Write the ratio: $\sin 30^\circ = \dfrac{x}{6}$
2. Replace with the exact value: $\dfrac{1}{2} = \dfrac{x}{6}$
3. Solve: $x = 6 \times \dfrac{1}{2} = 3$ cm