Download Sine, Cosine and Tangent

Trigonometric Functions in a Right Angled Triangle:
The right angle is shown by the little box in the corner. We
usually know another angle θ. And we give names to each side:
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Adjacent is adjacent (next to) to the angle θ
Opposite is opposite the angle θ
the longest side is the Hypotenuse
"Sine, Cosine and Tangent"
Trigonometry is good at find a missing side or angle in a triangle. The
special functions Sine, Cosine and Tangent will help us!
For any angle "θ":
Opposite is in front
of the angle
Adjacent is always next to
the angle
Sine Function: sin(θ) = Opposite / Hypotenuse
Cosine Function: cos(θ) = Adjacent / Hypotenuse
Tangent Function: tan(θ) = Opposite / Adjacent
(Sine, Cosine and Tangent are often abbreviated to:
sin, cos and tan.)
In Picture form:
Example: What is the sine, cosine and
tangent of 35°?
Using this triangle (lengths are
only to one decimal place):
sin(35°) = Opposite / Hypotenuse
= 2.8 / 4.9
= 0.57...
cos(35°) = Adjacent / Hypotenuse
= 4.0 / 4.9
= 0.82...
tan(35°) = Opposite / Adjacent
= 2.8 / 4.0
= 0.70...
Calculators have sin, cos and tan, let's see how to use them:
Example: What is the missing length here?
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We know the Hypotenuse: 20
We know the angle: 45 degrees
Sine is the ratio of Opposite / Hypotenuse
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Get a calculator, type in "45", then the "sin" key:
sin(45°) = 0.7071...
Now multiply by 20 (the Hypotenuse length):
Opposite length = 20 × 0.7071... = 14.14 (to 2 decimals)
How to remember? Think "SOHCAHTOA"! It works
like this:
SOH
Sine = Opposite / Hypotenuse
CAH
Cosine = Adjacent / Hypotenuse
TOA
Tangent = Opposite / Adjacent
Please remember it, it may help in an exam !
Why are these functions important?
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Because they let us work out angles when we know sides
And they let us work out sides when we know angles
Example: How deep is the ship from the bottom of
the ocean (seabed) "d" ?
We know:
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The cable makes a 39° angle with the seabed
The cable has a 30 meter length.
And we want to know "d" (the distance down).
Start with:
sin 39° = opposite/hypotenuse
sin 39° = d/30
Swap Sides:
d/30 = sin 39°
Use a calculator to find sin39°: d/30 = 0.6293…
Multiply both sides by 30:
d = 0.6293… x 30
d = 18.88 to 2 decimal places.
The depth "d" is 18.88 m
Inverse functions for sine, cosine, and
tangent.
The symbol for inverse sine is sin-1, inverse cosine is cos-1
and inverse tangent is tan-1.
The inverse functions are used when the angle is what we
need to find.
sin-1 (Opposite / Hypotenuse) = θ
cos-1 (Adjacent / Hypotenuse) = θ
tan-1 (Opposite / Adjacent) = θ
Example: Find the angle "a" between the seabed and the
anchor.
We know:
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The distance down is 18.88 m.
The cable's length is 30 m.
And we want to know the angle "a"
Calculator:
On the calculator you press one of
the following (depending on your
brand of calculator): either '2ndF
sin' or 'shift sin'.
Unit Circle
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What you just played with is the Unit Circle .
It is a circle with a radius of 1 with its center at 0.
Because the radius is 1, we can directly measure sine,
cosine and tangent.
Here we see the sine, cosine and tangent values:
https://www.mathsisfun.com/sine-cosine-tangent.html
Because the radius is 1, we can directly measure sine, cosine and tangent .
What happens when the angle, θ, is 0°? cos 0° = 1, sin 0° = 0 and tan 0° = 0
What happens when θ is 90°? cos 90° = 0, sin 90° = 1 and tan 90° is undefined
Degrees and Radians
Angles can be in Degrees or Radians . Here are some examples:
Angle
Degrees
Radians
Right Angle
90°
π/2
__ Straight Angle
180°
π
360°
2π
Full Rotation
What are 'radians' ?
One radian is the angle of an arc created by wrapping the radius of a
circle around its circumference.
In this diagram, the radius has been wrapped around the circumference
to create an angle of 1 radian. The pink lines show the radius being moved
from the inside of the circle to the outside:
The radius 'r' fits around the circumference of a circle exactly
2Π times. That is why the circumference of a circle is given by:
circumference = 2Πr
So there are 2Π radians in a complete circle, and
circle.
radians in a half
Converting radians to degrees:
To convert radians to degrees, we make use of the fact that Π radians
equals one half circle, or 180º.
This means that if we divide radians by Π, the answer is the number of
half circles. Multiplying this by 180º will tell us the answer in degrees.
So, to convert radians to degrees, multiply by 180/Π, like this:
Converting degrees to radians:
To convert degrees to radians, first find the number of half circles in the
answer by dividing by 180º. But each half circle equals Π radians, so
multiply the number of half circles by Π.
So, to convert degrees to radians, multiply by Π/180, like this:
The chart below shows the most commonly used values in
trigonometry.
Degrees
Radians Radians
(exact) (approx)
30°
π/6
0.524
45°
π/4
0.785
60°
π/3
1.047
90°
π/2
1.571
180°
π
3.142
270°
3π/2
4.712
360°
2π
6.283
To write radians in decimal form remember we use
Pi (π) = 3.14159265...
Repeating Pattern
Because the angle is rotating around and around the circle the Sine,
Cosine and Tangent functions repeat once every full rotation
When we need to calculate the function for an angle larger than a full
rotation of 2π (360°) we subtract as many full rotations as needed to
bring it back below 2π (360°):
Example: what is the cosine of 370°?
370° is greater than 360° so let us subtract 360°
370° − 360° = 10°
cos(370°) = cos(10°) = 0.985 (to 3 decimal places)
And when the angle is less than zero, just add full rotations.
Example: what is the sine of −3 radians?
−3 is less than 0 so let us add 2π radians
−3 + 2π = −3 + 6.283 = 3.283 radians
sin(−3) = sin(3.283) = −0.141 (to 3 decimal places)
Other Functions (Cotangent, Secant, Cosecant)
Similar to Sine, Cosine and Tangent, there are three
other trigonometric functions which are made by dividing one side by
another:
Cosecant Function:
csc(θ) = Hypotenuse / Opposite
Secant Function:
sec(θ) = Hypotenuse / Adjacent
Cotangent Function:
cot(θ) = Adjacent / Opposite
The Law of Sines (It works for any triangle)
The Law of Sines (or Sine Rule) is very useful for solving triangles:
And it says that:
When we divide side a by the sine of angle A
it is equal to side b divided by the sine of angle B,
and also equal to side c divided by the sine of angle C
Example:
Example: Calculate side "c"
Law of Sines: a/sin A = b/sin B = c/sin C
Put in the values we know: a/sin A = 7/sin(35°) = c/sin(105°)
Ignore a/sin A (not useful to us): 7/sin(35°) = c/sin(105°)
Now we use our algebra skills to rearrange and solve:
Swap sides: c/sin(105°) = 7/sin(35°)
Multiply both sides by sin(105°): c = ( 7 / sin(35°) ) × sin(105°)
Calculate: c = ( 7 / 0.574... ) × 0.966...
Calculate: c = 11.8 (to 1 decimal place)
Example: Calculate angle B
Start with: sin A / a = sin B / b = sin C / c
Put in the values we know: sin A / a = sin B / 4.7 = sin(63º) / 5.5
Ignore "sin A / a": sin B / 4.7 = sin(63º) / 5.5
Multiply both sides by 4.7: sin B = (sin63º/5.5) × 4.7
Calculate: sin B = 0.7614...
Inverse Sine: B = sin-1(0.7614...)
B = 49.6º
Example: Calculate angle R
The first thing to notice is that this triangle has different labels: PQR
instead of ABC. But that's OK. We just use P,Q and R instead of A, B
and C in The Law of Sines.
Start with: sin R / r = sin Q / q
Put in the values we know: sin R / 41 = sin(39º)/28
Multiply both sides by 41: sin R = (sin39º/28) × 41
Calculate: sin R = 0.9215...
Inverse Sine: R = sin-1(0.9215...)
R = 67.1º
But wait! There's another angle that also has a sine equal to 0.9215...
The calculator won't tell you this but sin(112.9º) is also equal to
0.9215...
So, how do we discover the value 112.9º?
Easy ... take 67.1º away from 180°, like this:
180° - 67.1° = 112.9°
So there are two possible answers for R: 67.1º and 112.9º:
Both are possible! Each one has the 39º angle, and sides of 41 and 28.
So, always check to see whether the alternative answer makes sense.
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... sometimes it will (like above) and there are two solutions
... sometimes it won't (see below) and there is one solution
We looked at this triangle before.
As you can see, you can try swinging the "5.5" line around, but no other
solution makes sense. So this has only one solution.
The Law of Cosines
The Law of Cosines (also called the Cosine Rule) is very useful for
solving triangles:
It works for any triangle: a, b and c are sides.
C is the angle opposite side c
Versions for finding sides a and b
But it is easier to remember the "c2=" form and
change the letters as needed!
Example: How long is side "c" ... ?
We know angle C = 37º, a = 8 and b = 11
The Law of 2
c = a2 + b2 − 2ab cos(C)
Cosines says:
Substitute the values 2
c = 82 + 112 − 2 × 8 × 11 × cos(37º)
we know:
Do some calculations: c2 = 64 + 121 − 176 × 0.798…
More calculations: c2 = 44.44...
Take the square root: c = √44.44 = 6.67 to 2 decimal places
Answer: c = 6.67
Example: Find the distance "z"
The letters are different! But that doesn't matter. We can
easily substitute x for a, y for b and z for c
Start with:
x for a,
y for b
z for c
substitute the data:
Calculate:
c2 = a2 + b2 − 2ab cos(C)
z2 = x2 + y2 − 2xy cos(Z)
z2 = 9.42 + 6.52 − 2×9.4×6.5×cos(131º)
z2 = 88.36 + 42.25 − 122.2 × (−0.656...)
z2 = 130.61 + 80.17...
z2 = 210.78...
z = √210.78... = 14.5 to 1 decimal
place.
Answer: z = 14.5
Did you notice that cos(131º) is negative and this changes the last sign
in the calculation to + (plus)? The cosine of an obtuse angle is always
negative (see Unit Circle ).
How to Remember
How can you remember the formula?
Well, it helps to know it's the Pythagoras Theorem with something
extra so it works for all triangles:
Pythagoras 2
a + b2 = c2
Theorem:
(only for Right-Angled
Triangles)
Law of Cosines: a2 + b2 − 2ab cos(C) = c2 (for all triangles)
So, to remember it:
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think "abc": a2 + b2 = c2,
then a 2nd "abc": 2ab cos(C),
and put them together: a2 + b2 − 2ab cos(C) = c2
When to Use
The Law of Cosines is useful for finding:
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the third side of a triangle when we know two sides and the angle between
them (like the example above)
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the angles of a triangle when we know all three sides (as in the following
example)
Example: What is Angle "C" ...?
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The side of length "8" is opposite angle C, so it is side c.
The other two sides are a and b.
Now let us put what we know into The Law of Cosines:
Start with: c2 = a2 + b2 − 2ab cos(C)
Put in a, b and c: 82 = 92 + 52 − 2 × 9 × 5 × cos(C)
Calculate: 64 = 81 + 25 − 90 × cos(C)
Now we use our algebra skills to rearrange and solve:
Subtract 25 from both sides: 39 = 81 − 90 × cos(C)
Subtract 81 from both sides: −42 = −90 × cos(C)
Swap sides: −90 × cos(C) = −42
Divide both sides by −90: cos(C) = 42/90
Inverse cosine: C = cos-1(42/90)
Calculator: C = 62.2° (to 1 decimal place)
In Other Forms
Easier Version For Angles
We just saw how to find an angle when we know three sides. It took
quite a few steps, so it is easier to use the "direct" formula (which is
just a rearrangement of the c2 = a2 + b2 − 2ab cos(C) formula):
Example: Find Angle "C" Using The Law of Cosines (angle
version)
In this triangle we know the three sides:
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a=8
b=6
c=7
Use The Law of Cosines (angle version) to find angle C :
cos C = (a2 + b2 − c2)/2ab
= (82 + 62 − 72)/2×8×6
= (64 + 36 − 49)/96
= 51/96
= 0.53125
C = cos-1(0.53125) = 57.9° to one decimal
place