All The Trig That You Need To Know
All The Trig That You Need To Know
All The Trig That You Need To Know
Given angle θ in standard position and point P(x, y) on the terminal side:
y x y
sin θ = cos θ = tan θ =
r r x
x r r
cot θ = sec θ = csc θ =
y x y
MEASURING ANGLES
π
To covert degrees to radians: RADIANS = (DEGREES) X
180
180
To convert radians to degrees: DEGREES = (RADIANS) X
π
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TRIG FUNCTIONS OF IMPORTANT ANGLES
The following table shows the trigonometric functions of certain important first
quadrant angles:
REFERENCE ANGLE
The reference angle for a given angle, θ, is the acute angle between the terminal
side of θ and the x-axis.
The trig functions of an angle and its reference angle are numerically equal. The
signs, however, may be positive or negative depending on the quadrant in which
angle θ lies. (see Chipmunks)
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GRAPHS OF THE TRIGONOMETRIC FUNCTIONS
The graphs of the six basic trig functions are shown below:
AMPLITUDE
The amplitude of each of the graphs above is “1.” For sine and cosine it’s
the distance from the zero point to the top of the graph. For tangent and
cotangent, it’s the value of the function halfway between the zero point and the
asymptote. For secant and cosecant, it’s the distance from the x-axis to the low
point on the graph.
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FREQUENCY AND PERIOD
The frequency can be changed if you multiply the argument of the function
(the x) by a number.
usual _ period
T=
frequency
360° 2π
For sine, cosine, secant and cosecant, T = or T = , where f is the
f f
frequency.
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PHASE SHIFT
As was the case with other functions we studied, we can shift the graph left or
right by adding or subtracting a number from the “x.”
π
y = sin x − is a sine graph shifted y = sec (x + 60°) is a secant graph
4
an amount π/4 to the right. shifted 60° to the left.
The formulas below show the basic relationships between the six
trigonometric functions of any angle.
1 sin θ
csc θ = tan θ =
sin θ cos θ
1 cos θ
sec θ = cot θ =
cos θ sin θ
1
cot θ =
tan θ
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COFUNCTION PROPERTIES
When two angles are complementary (that is, they add up to 90° or to π/2, such
π 3π
as 35° and 55°, or and , or even 0° and 90°, or 110° and –20°), their
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cofunctions are equal, that is, the sine of one equals the cosine of the other, the
tangent of one equals the cotangent of the other, or the secant of one equals the
cosecant of the other.
π 3π
sin 35° = cos 55°; cot = tan ; sec(– 20°) = csc 110°
8 8
Angles θ and (90° – θ) are complementary as are angles x and (π/2 – x), so
First, make sure your calculator is set to the proper MODE. If you are
using angles in degrees, select Degree mode, or, select Radian mode for
angles measured in radians.
You can not directly find the cotangent, secant, or cosecant of an angle.
You must use the reciprocal identities.
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DEGREES, MINUTES, AND SECONDS
Your calculator does not handle degrees, minutes, and seconds well.
Instead you should convert the angle to degrees only.
If you know that sin θ = .3761, you find the angle by asking the calculator
for the inverse sine of .3761:
REMEMBER: sin θ = .3761 has two answers between 0° and 360°. The
calculator gives you the first quadrant angle (which is also the reference angle).
There is another answer in the second quadrant at 180° – 22°5’32.337” which
equals 157°54’27.663”.
If you know that sec θ = 3.29, you must first rewrite the problem as
cos θ = 1/3.29. Then perform an operation similar to the one above:
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WHEN THE KNOWN FUNCTION VALUE IS NEGATIVE
-1
The command sin (negative number) gives you an answer between 0° and –
90° (fourth quadrant, but not between 0° and 360°). The correct fourth quadrant
answer is 360° + sin -1(negative number). The correct third quadrant answer is
180° – sin -1(negative number).
-1
The command tan (negative number) gives you an answer between 0° and –
90° (fourth quadrant, but not between 0° and 360°). The correct fourth quadrant
-1
answer is 360° + tan (negative number). The correct second quadrant answer
-1
is 180° + tan (negative number).
-1
The command cos (negative number) gives you an answer between 90° and
180°, the correct second quadrant answer. The correct third quadrant answer is
-1
360° – cos (negative number).
-1
sin (- .5958) = – 36.569°
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RIGHT TRIANGLE TRIGONOMETRY
Standard labeling for a right triangle has angles A, B, and C, with ∠C being the
right angle. Sides a, b, and c, are opposite angles A, B, and C, respectively. The
hypotenuse of a right triangle is the side opposite the right angle. For either of the
acute angles, the side which is not opposite the angle and which is not the
hypotenuse is called the adjacent side.
opposite
sin θ =
hypotenuse
adjacent
cos θ =
hypotenuse
opposite
tan θ =
adjacent
Solving a right triangle means finding all three sides and all three angles.
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OBLIQUE TRIANGLES
An oblique triangle (a triangle with no right angle) can be solved if you know
If you know two sides and the angle between (a-C-b, a-B-c, or b-A-c), the third
side can be found using the LAW OF COSINES:
2 2 2
c = a + b – 2ab cos C
2 2 2
or b = a + c – 2ac cos B
2 2 2
or a = b + c – 2bc cos A
Once you know all three sides, you can find the angles:
When you know all three sides of any triangle, the LAW OF COSINES can be
used to find the cosines of each of the three angles:
b2 + c 2 − a2 a2 + c 2 − b2 a2 + b2 − c 2
cos A = cos B = cos C =
2bc 2ac 2ab
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THE CASE OF ASA OR SAA
First, if you know any two angles, calculate the third. Now you know all three
angles and you can apply the LAW OF SINES to find the missing two sides. The
Law of Sines states:
a b a c b c
= or = or =
sin A sin B sin A sin C sin B sin C
a b c
sometimes written: = =
sin A sin B sin C
In the case of two sides with the angle NOT between them (such as a-b-A or
a-c-C or a-b-B) you can use the Law of Sines to calculate the sine of the second
angle. This can be a tricky proposition, because if we calculate, for example,
sin B = 0.9563
our calculator says B = 73°
but we must remember sin 107° = 0.9563 also.
It is possible, depending on the size of the other angle in the problem (we can’t
go over 180° total) that both answers may be correct, and there are two
different triangles that fit the given information. When this happens (but not
always---but you always have to check for it) we have what is called the
ambiguous case, and both answers must be given.
It is also possible when using the Law of Sines that in calculating the sine of the
second angle we may get something like
sin B = 1.1208
We know that for any angle B, – 1 < sin B < 1. sin B = 1.1208 is an impossible
situation, and there is no triangle that fits the given information.
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OTHER TRIG FORMULAS THAT YOU DON’T NEED TO KNOW*
*But that future math courses may expect you to know
If you know the trig functions of angles α and β, you can find the trig functions of
the combined angle (α + β) or (α – β) from the formulas:
sin(α + β) = sin α cos β + cos α sin β sin(α − β) = sin α cos β – cos α sin β
cos(α + β) = cos α cos β – sin α sin β cos(α − β) = cos α cos β + sin α sin β
2 tan x
tan 2x =
1 − tan 2 x
HALF-ANGLE FORMULAS
1 1− cos x 1 1+ cos x
sin x= ± cos x= ±
2 2 2 2
1 1 − cos x
tan x= ±
2 1 + cos x
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OPPOSITE ANGLES
PRODUCT TO SUM
1 1
sin ax sin bx = cos(a – b)x – cos(a + b)x
2 2
1 1
cos ax cos bx = cos(a – b)x + cos(a + b)x
2 2
1 1
sin ax cos bx = sin(a + b)x + sin(a – b)x
2 2
SUM TO PRODUCT
a+b a−b
sin ax + sin bx = 2 sin x cos x
2 2
a+b a−b
cos ax + cos bx = 2 cos x cos x
2 2
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INVERSE TRIG FUNCTIONS
The expression arcsin x or sin – 1 x is read “inverse sine of x” or “the angle whose
sine is x.”
π π
Example: arcsin ½ = 30° or arctan 1 = 45° or
6 4
Since this expression refers to a “function” of x, there can be only one angle
whose sine is x (although we known there are an infinite number of angles whose
sine is, say, 1/2).
π π
– < arcsin x <
2 2
π
0 < arcsec x < or
2
3π
π < arcsec x <
2
π
– π < arccsc x < – or
2
π
0 < arccsc x <
2
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The graphs of the inverse functions are shown below:
Y = arcsin x y = arcos x
y = artan x y = arccot x
y = arcsec x y = arccsc x
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Because sin x and arcsin x are inverse functions, the composition of the two
functions should return the value of x, that is,
3π 3π
So, arctan(tan 17°) = 17°, and arcos(cos )= .
4 4
This means arcsin(sin 200°) = – 20°. [The angle <in range> whose sine is the
same as the sine of 200° <not in range> is – 20°.]
3π 5π 3π
Likewise arcsec( sec )= . [ is not in the range of arcsec x.]
4 4 4
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