True or False? The hypotenuse is the side opposite the angle you are working with.

False. The hypotenuse is fixed by the right angle , not by whichever angle you happen to be using. It stays the longest side, opposite the right angle, whichever of the other two angles you choose.

How do you find the angle between a line and a plane?

Drop a perpendicular from a point on the line down onto the plane, which creates a right-angled triangle. The angle wanted is the one between the line and its shadow on the plane, and SOHCAHTOA then finds it.

IGCSEFurther MathsEdexcelFlashcardsGeometry & TrigonometryTrigonometric Ratios

Trigonometric Ratios (Edexcel IGCSE Further Pure Maths): Flashcards

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FrontBasic Trigonometry
Complete Pythagoras' theorem for a right-angled triangle with hypotenuse $c$ :
$c^{2} =______+______$
FrontBasic Trigonometry
What are the three SOHCAHTOA ratios?
BackBasic Trigonometry
Measured against a chosen angle $θ$ :
$\sin θ = \frac{opposite}{hypotenuse}$
$\cos θ = \frac{adjacent}{hypotenuse}$
$\tan θ = \frac{opposite}{adjacent}$
They work in a right-angled triangle only.

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Cards in this collection (24)

Complete Pythagoras' theorem for a right-angled triangle with hypotenuse $c$ :
$c^{2} =______+______$
The completed theorem is:
$c^{2} = a^{2} + b^{2}$
It holds only in a right-angled triangle, and $c$ must be the hypotenuse.
What are the three SOHCAHTOA ratios?
Measured against a chosen angle $θ$ :
- $\sin θ = \frac{opposite}{hypotenuse}$
- $\cos θ = \frac{adjacent}{hypotenuse}$
- $\tan θ = \frac{opposite}{adjacent}$
They work in a right-angled triangle only.
How do you label the sides of a right-angled triangle before using SOHCAHTOA?
Pick your angle $θ$ first: the side facing it is the opposite, and the side between $θ$ and the right angle is the adjacent.
Getting those two the wrong way round is what turns a $\sin$ question into a $\cos$ one.
How can Pythagoras' theorem show that a triangle is right-angled?
Test whether $c^{2} = a^{2} + b^{2}$ holds for the three side lengths, taking $c$ as the longest.
If it does, the triangle is right-angled; if it does not, the triangle is not.
True or False?
The hypotenuse is the side opposite the angle you are working with.
False.
The hypotenuse is fixed by the right angle, not by whichever angle you happen to be using.
It stays the longest side, opposite the right angle, whichever of the other two angles you choose.
What is the 3D version of Pythagoras' theorem?
$l^{2} = x^{2} + y^{2} + z^{2}$ , where $x$ , $y$ and $z$ are the distances in the three directions.
In practice it is usually easier to break the solid into two right-angled triangles and apply the ordinary theorem twice.
How do you find the angle between a line and a plane?
Drop a perpendicular from a point on the line down onto the plane, which creates a right-angled triangle.
The angle wanted is the one between the line and its shadow on the plane, and SOHCAHTOA then finds it.
Define unit circle.
The unit circle is the circle of radius $1$ centred on the origin.
It extends $\sin$ , $\cos$ and $\tan$ to angles of any size, including those beyond $360 °$ and negative ones.
On the unit circle, what do the $x$ and $y$ coordinates give?
The $x$ coordinate is $\cos θ$ and the $y$ coordinate is $\sin θ$ , with $θ$ measured from the positive $x$ -axis.
Dividing $y$ by $x$ gives $\tan θ$ , which is also the gradient of the line from the origin to that point.
Which way round are positive and negative angles measured?
Anticlockwise from the positive $x$ -axis for a positive angle, and clockwise for a negative one.
Going past a full turn simply continues, so $420 °$ is one complete revolution and then another $60 °$ .
What does the CAST diagram tell you?
Which function is positive in each quadrant, reading anticlockwise from the fourth: Cosine, All, Sine, Tangent.
So in the second quadrant only $\sin θ$ is positive, and in the third only $\tan θ$ is.
True or False?
$\sin 210 °$ and $\sin 330 °$ have the same value.
True.
Both equal $- \frac{1}{2}$ : each is a $30 °$ angle measured from the horizontal, and sine is negative in the third and fourth quadrants alike.
The values match even though the two angles sit in different quadrants.
Complete the exact values of $\sin$ for these three angles:
$\sin 30 ° =______, \sin 45 ° = \frac{\sqrt{2}}{2}, \sin 60 ° =______$
The completed values are:
$\sin 30 ° = \frac{1}{2}, \sin 45 ° = \frac{\sqrt{2}}{2}, \sin 60 ° = \frac{\sqrt{3}}{2}$
Written as $\frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}$ the pattern is plain, and the cosine values run the same way backwards.
What are the exact values of $\tan 30 °$ , $\tan 45 °$ and $\tan 60 °$ ?
They are $\frac{1}{\sqrt{3}}$ , $1$ and $\sqrt{3}$ .
Each follows from $\tan θ = \frac{\sin θ}{\cos θ}$ , so they can be rebuilt rather than memorised separately.
Which triangles let you derive the exact values for $30 °$ , $45 °$ and $60 °$ ?
An equilateral triangle of side $2$ , cut in half down the middle, gives $30 °$ and $60 °$ .
A right-angled isosceles triangle with two sides of $1$ gives $45 °$ .
How do you get $\sin 150 °$ from $\sin 30 °$ ?
Draw the same $30 °$ angle up from the horizontal in the second quadrant, where it becomes $150 °$ .
Sine is positive in that quadrant, so $\sin 150 ° = \sin 30 ° = \frac{1}{2}$ .
Complete the periods of the three trigonometric graphs:
$\sin x and \cos x repeat every______°, but \tan x repeats every______°$
The completed periods are:
$\sin x and \cos x repeat every 360 °, but \tan x repeats every 180 °$
In radians those are $2 π$ and $π$ , and the shorter period is why $\tan$ gives twice as many solutions in the same interval.
What is the range of $\sin x$ and $\cos x$ ?
It is $- 1 \leq y \leq 1$ , so neither can ever rise above $1$ or fall below $- 1$ .
That is why an equation such as $\sin x = 2$ has no solutions at all.
Where is $\tan x$ undefined, and what is its range?
It is undefined at $\pm 90 °$ , $\pm 270 °$ and so on, where it has vertical asymptotes.
Its range is every real number, so unlike $\sin$ and $\cos$ it is not bounded above or below.
How are the graphs of $\sin x$ and $\cos x$ related?
They are translations of one another: shifting $\sin x$ left by $90 °$ gives $\cos x$ .
That is why $\sin x$ passes through the origin while $\cos x$ passes through $(0, 1)$ .
True or False?
A calculator gives you every solution of $\sin x = 0.5$ in a given interval.
False.
The inverse function returns only the primary value, one solution out of infinitely many.
The symmetry of the graph is what you then use to find the rest inside the required interval.
How does a sketch tell you how many solutions $\sin x = k$ has?
Draw the horizontal line $y = k$ across the sketch and count where it crosses the curve.
The number of crossings inside the given interval is the number of solutions you should finish with.
What two symmetry properties does the graph of $\sin x$ have?
It has rotational symmetry about the origin, so $\sin (- x) = - \sin x$ .
It is also symmetric about $x = 90 °$ , so $\sin x = \sin (180 ° - x)$ .
What two symmetry properties does the graph of $\cos x$ have?
It is symmetric about the vertical axis, so $\cos (- x) = \cos x$ .
It also repeats about a full turn, so $\cos x = \cos (360 ° - x)$ .

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