How do you identify the opposite and adjacent sides in a right-angled triangle?

They are named relative to the angle you are using : the opposite is across from it, and the adjacent lies between it and the right angle. The hypotenuse is the only side that never changes, being the longest and always opposite the right angle.

True or False? SOH CAH TOA can be used on any triangle.

False. It works only in a right-angled triangle, because the ratios are defined against the hypotenuse. Any other triangle needs the sine rule or the cosine rule instead.

How can you check that a side length you have found is sensible?

Compare it against the sides you already know. The hypotenuse has to be the longest side, so an answer longer than it, or a hypotenuse shorter than one of the others, must be wrong.

When do you use the sine rule ?

When the question involves opposite pairs of a side and its angle. You need one complete pair, plus one other side or angle to work from.

When do you use the cosine rule ?

When you are given an angle between two known sides, or all three sides. Neither of those situations gives you a complete opposite pair to work with.

What is the first thing to check before choosing a rule for a triangle?

Whether the triangle is right-angled . If it is, the simple trigonometric ratios will do the job far more quickly.

True or False? The sine rule and cosine rule work on right-angled triangles as well.

True. Both hold for any triangle, right-angled or not. For a right-angled triangle the simple ratios are quicker, which is why they are used instead, not because the rules fail.

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Basic Trigonometry (Edexcel A Level Maths: Pure): Flashcards

Exam code: 9MA0

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FrontDefinitions of Trigonometric Functions
Define the unit circle.

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Cards in this collection (19)

Define the unit circle.
A circle of radius $1$ centred on the origin, used to define the trigonometric functions.
Angles on it are measured anticlockwise from the positive $x$ -axis.
What does the unit circle let you do that a right-angled triangle cannot?
Find $\sin$ , $\cos$ and $\tan$ for angles greater than $90 °$ .
No right-angled triangle can contain an angle that big, so the ratio definitions on their own do not reach that far.
On the unit circle, the point at angle $θ$ has coordinates:
$(______,______)$
$(\cos θ, \sin θ)$
The $x$ -coordinate is the cosine and the $y$ -coordinate is the sine, which is the order most often reversed.
Where do $\sin$ , $\cos$ and $\tan$ come from originally?
From ratios of side lengths in a right-angled triangle.
The word trigonometry itself is from the Greek for "triangle" and "measure".
True or False?
On the unit circle, $\cos θ$ can be greater than $1$ .
False.
Every point on the circle is exactly $1$ unit from the origin, so neither of its coordinates can exceed $1$ .
That is why $\sin θ$ and $\cos θ$ always lie between $- 1$ and $1$ .
Why does the unit circle have radius $1$ rather than any other value?
Because the hypotenuse of the triangle inside it is then $1$ , so dividing by it changes nothing.
The two ratios therefore become simply the two coordinates of the point.
SOH CAH TOA: $\sin θ = \frac{O}{H}$ , $\cos θ = \frac{______}{H}$ , $\tan θ = \frac{O}{______}$ .
$\sin θ = \frac{O}{H}$ , $\cos θ = \frac{A}{H}$ , $\tan θ = \frac{O}{A}$ .
Here $O$ , $A$ and $H$ are the opposite, adjacent and hypotenuse.
How do you identify the opposite and adjacent sides in a right-angled triangle?
They are named relative to the angle you are using: the opposite is across from it, and the adjacent lies between it and the right angle.
The hypotenuse is the only side that never changes, being the longest and always opposite the right angle.
How do you find a missing angle in a right-angled triangle?
Work out the ratio of the two sides you know, then apply the inverse function, $\sin^{- 1}$ , $\cos^{- 1}$ or $\tan^{- 1}$ .
Without the inverse you get the ratio back rather than the angle itself.
True or False?
SOH CAH TOA can be used on any triangle.
False.
It works only in a right-angled triangle, because the ratios are defined against the hypotenuse.
Any other triangle needs the sine rule or the cosine rule instead.
How can you check that a side length you have found is sensible?
Compare it against the sides you already know.
The hypotenuse has to be the longest side, so an answer longer than it, or a hypotenuse shorter than one of the others, must be wrong.
The cosine rule is:
$a^{2} = b^{2} +______- 2 b c \cos______$
$a^{2} = b^{2} + c^{2} - 2 b c \cos A$
The angle $A$ is the one opposite the side $a$ , which is what makes the formula work.
When do you use the sine rule?
When the question involves opposite pairs of a side and its angle.
You need one complete pair, plus one other side or angle to work from.
When do you use the cosine rule?
When you are given an angle between two known sides, or all three sides.
Neither of those situations gives you a complete opposite pair to work with.
Why might the angle your calculator gives from the sine rule not be the one you want?
Because it always returns the acute angle, and the triangle may need the obtuse one instead.
Subtracting from $180 °$ gives the other possibility, and the shape of the triangle decides which fits.
What is the formula for the area of any triangle?
$Area = \frac{1}{2} a b \sin C$
The angle $C$ has to be the one between the two sides $a$ and $b$ .
What is the first thing to check before choosing a rule for a triangle?
Whether the triangle is right-angled.
If it is, the simple trigonometric ratios will do the job far more quickly.
Two angles of a triangle are given. How does that help?
The third angle follows immediately, since the three add to $180 °$ .
That often turns a problem with no usable opposite pair into one that has.
True or False?
The sine rule and cosine rule work on right-angled triangles as well.
True.
Both hold for any triangle, right-angled or not.
For a right-angled triangle the simple ratios are quicker, which is why they are used instead, not because the rules fail.

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