Pythagoras & Trigonometry Flashcards

All 29 cards in this deck

What is Pythagoras' theorem?

$a^2 + b^2 = c^2$, where $c$ is the hypotenuse and $a$, $b$ 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: $\sqrt{100} = 10$ cm

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

e.g. hypotenuse $13$ cm, one shorter side $5$ cm

1. Square both known sides: $13^2 = 169$, $5^2 = 25$
2. Subtract from the hypotenuse squared: $169 - 25 = 144$
3. Square root: $\sqrt{144} = 12$ cm

True or false? To find a shorter side of a right-angled triangle you add the squares of the other two sides.

False. You subtract: shorter side $= \sqrt{\text{hyp}^2 - \text{other side}^2}$.

How do you test whether a triangle with three given sides is right-angled?

e.g. sides $5$ cm, $12$ cm, $13$ cm

Check whether the squares of the two shorter sides add to the square of the longest side: $25 + 144 = 169 = 13^2$, so it is right-angled.

A straight ladder leans against a vertical wall. In the right-angled triangle formed by the ladder, the wall and the ground, which side is the hypotenuse?

The ladder. It is opposite the right angle between the wall and the ground, so it is the longest side.

What are the three trigonometric ratios for a right-angled triangle?

$\sin\theta = \frac{\text{opp}}{\text{hyp}}$, $\cos\theta = \frac{\text{adj}}{\text{hyp}}$, $\tan\theta = \frac{\text{opp}}{\text{adj}}$ (SOH CAH TOA).

How are the three sides of a right-angled triangle labelled relative to a marked angle $\theta$?

Hypotenuse: opposite the right angle (longest side). Opposite: across from $\theta$. Adjacent: between $\theta$ and the right angle.

Which trigonometric ratio do you use when you know the hypotenuse and want the side opposite the given angle?

e.g. hypotenuse $178$ mm, angle $34^\circ$, opposite side $x$

Sine: $\sin 34^\circ = \frac{x}{178}$, so $x = 178 \times \sin 34^\circ$.

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

e.g. angle $56^\circ$, adjacent side $12$ cm, find the opposite side $BC$

1. Label the sides for $56^\circ$: $12$ is adjacent, $BC$ is opposite
2. Choose the ratio: $\tan 56^\circ = \frac{BC}{12}$
3. Rearrange and work out: $BC = 12 \times \tan 56^\circ = 17.8$ cm (1 d.p.)

What do you do when the unknown side is the denominator of the trig ratio?

e.g. $\sin 30^\circ = \frac{5}{x}$

Divide the known side by the ratio: $x = 5 \div \sin 30^\circ = 10$.

What are the steps to find the area of a right-angled triangle when you know the base and one acute angle?

e.g. base $10$ cm, angle $45^\circ$ between base and hypotenuse

1. Set up the ratio for the height: $\tan 45^\circ = \frac{h}{10}$
2. Find the height: $h = 10 \times \tan 45^\circ = 10$ cm
3. Area $= \frac{1}{2} \times 10 \times 10 = 50$ cm$^2$

How do you find an angle once you know the value of a trigonometric ratio?

e.g. $\cos x = \frac{15}{18}$

Use the inverse function: $x = \cos^{-1}\left(\frac{15}{18}\right) = 33.6^\circ$ (1 d.p.).

What are the steps to find an angle in a right-angled triangle from two known sides?

e.g. side adjacent to angle $ABC$ is $7$ cm, hypotenuse is $11$ cm

1. Label the known sides for angle $ABC$: $7$ adjacent, $11$ hypotenuse
2. Write the ratio: $\cos ABC = \frac{7}{11}$
3. Use the inverse: $ABC = \cos^{-1}\left(\frac{7}{11}\right) = 50.5^\circ$ (1 d.p.)

What does $\sin^{-1}$ do?

It gives the angle whose sine is that value (the inverse sine function).

True or false? $\sin^{-1} x$ means $\frac{1}{\sin x}$.

False. $\sin^{-1}x$ is the inverse sine — the angle whose sine is $x$.

As an acute angle gets larger, what happens to its cosine?

It decreases, from $\cos 0^\circ = 1$ down to $\cos 90^\circ = 0$.

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 elevation of B from A is equal to the angle of depression of A from B.

True. The two horizontal lines are parallel, so the angles are alternate angles.

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

e.g. angle of elevation $30^\circ$ from a point $20$ m from the base of a tower

1. Sketch the right-angled triangle: $20$ m adjacent, height $h$ opposite the $30^\circ$
2. Choose the ratio: $\tan 30^\circ = \frac{h}{20}$
3. Work it out: $h = 20 \times \tan 30^\circ = 11.5$ m (1 d.p.)

What are the exact values of $\sin 30^\circ$, $\cos 30^\circ$ and $\tan 30^\circ$?

$\sin 30^\circ = \frac{1}{2}$, $\cos 30^\circ = \frac{\sqrt{3}}{2}$, $\tan 30^\circ = \frac{1}{\sqrt{3}}$

What are the exact values of $\sin 45^\circ$, $\cos 45^\circ$ and $\tan 45^\circ$?

$\sin 45^\circ = \cos 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$, $\tan 45^\circ = 1$

What are the exact values of $\sin 60^\circ$, $\cos 60^\circ$ and $\tan 60^\circ$?

$\sin 60^\circ = \frac{\sqrt{3}}{2}$, $\cos 60^\circ = \frac{1}{2}$, $\tan 60^\circ = \sqrt{3}$

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

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

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

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

True or false? $\sin 30^\circ = \cos 60^\circ$.

True. Both are exactly $\frac{1}{2}$.

What is the exact value of $\tan 45^\circ$?

$1$

In a right-angled triangle with a $45^\circ$ angle the opposite and adjacent sides are equal.

True or false? $\cos 0^\circ = 0$.

False.

$\cos 0^\circ = 1$; it is $\sin 0^\circ$ that equals $0$.