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TrigonometryClass:IX
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Trigonometry
Trigonometry is the branch of mathematics which deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degrees.
.
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------------------------------------------------
Trigonometry specifically deals with
the relationships between the
sides and the angles of triangles.
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In trigonometry, the ratio we are
talking about is the comparison of
the sides of a RIGHT ANGLED
TRIANGLE. Two things MUST BE understood:
1. This is the hypotenuse.
.
2. This is 90°
Now that we agree about the hypotenuse and right angle, there are only 4 things left; the 2 other angles and the 2 other sides.
A.
Opposite side
Ad
jace
nt
sid
e
Hypotenuse
Remember we use the right angle
√
θ this is the symbol for an unknown angle
measure.
It’s name is ‘Theta’.
One more thing…
Trigonometric Ratios
Name
“say” Sine Cosine tangent
Abbreviation
Sin Cos Tan
Ratio of an
angle
measure
Sinθ = opposite side
hypotenuse
cosθ = adjacent side
hypotenuse
tanθ =opposite side
adjacent side
One more
time…
Here are the
ratios:
sinθ = opposite side
hypotenuse
cosθ = adjacent side
hypotenuse
tanθ =opposite side
adjacent side
A trigonometric equation is an equation that involves
at least one trigonometric function of a variable. The
equation is a trigonometric identity if it is true for all
values of the variable for which both sides of the
equation are defined.
Trigonometric Identities
Prove that tan sin
cos.
y
x
y
r
x
r
y
r
r
x
y
x
L.S. = R.S.5.4.2
Recall the basic
trig identities:
sin y
r
cos x
r
tan y
x
5.4.3
Trigonometric Identities
Quotient Identities
tan sin
coscot
cos
sin
Reciprocal Identities
sin 1
csccos
1
sectan
1
cot
Pythagorean Identities
sin2 + cos2 = 1 tan2 + 1 = sec2 cot2 + 1 = csc2
sin2 = 1 - cos2
cos2 = 1 - sin2
tan2 = sec2 - 1 cot2 = csc2 - 1
sinx x sinx = sin2x
cos 1
cos
cos2
cos
1
cos
cos2 1
cos
sinA cos A 2
sin2
A 2sinAcos A cos2
A
12sinAcosA
cosA
sinA1
sinA
cosA
sinA
sinA
1
= cosA
Trigonometric Identities [cont’d]
5.4.4
Identities can be used to simplify trigonometric expressions.
Simplifying Trigonometric Expressions
cos sin tan
cos sin
sin
cos
cos
sin2
cos
cos 2 sin2
cos
1
cos
sec
a)
Simplify.
b)cot2
1 sin2
cos 2
sin2 cos
2
1
1
sin2
csc2
5.4.5
cos2
sin2
1
cos2
5.4.6
Simplifing Trigonometric Expressions
c) (1 + tan x)2 - 2 sin x sec x
1 2 tanx tan2x 2
sinx
cosx
1 tan2x 2tanx 2tanx
sec2x
d)cscx
tanx cotx
1
sinx
sinx
cosx
cosx
sinx
1
sinx
sin2x cos
2x
sinxcos x
1
sinx
sinx cos x
1
cos x
1
sinx
1
sinx cosx
(1 tanx)2
2 sinx1
cosx
5.4.7
Proving an Identity
Steps in Proving Identities:
1. Start with the more complex side of the identity and work
with it exclusively to transform the expression into the
simpler side of the identity.
2. Look for algebraic simplifications:
• Do any multiplying , factoring, or squaring which is
obvious in the expression.
• Reduce two terms to one, either add two terms or
factor so that you may reduce.
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3. Look for trigonometric simplifications
• Look for familiar trig relationships :
• If the expression contains squared terms
• , think of the Pythagorean Identities.
Transform each term to sine or cosine, if the
expression cannot be simplified easily using
other ratios.
5.4.8
Proving an Identity
Prove the following:
a) sec x(1 + cos x) = 1 + sec x
= sec x + sec x cos x
= sec x + 1
1 + sec x
L.S. = R.S.
b) sec x = tan x csc x
sinx
cos x
1
sinx
1
cosx
secx
secx
L.S. = R.S.
c) tan x sin x + cos x = sec x
sinx
cosx
sinx
1 cosx
sin2 x cos 2 x
cos x
1
cosx
secx
secx
L.S. = R.S.
d) sin4x - cos4x = 1 - 2cos2 x
= (sin2x - cos2x)(sin2x + cos2x)
= (1 - cos2x - cos2x)
= 1 - 2cos2x
L.S. = R.S.
1 - 2cos2x
e)
1
1 cosx
1
1 cosx 2 csc
2x
(1 cosx) (1 cosx)
(1 cosx)(1 cosx)
2
(1 cos2
x)
2
sin2x
2csc2x
2csc2x
L.S. = R.S.
Proving an Identity
5.4.9
Proving an Identity
5.4.10
f)
cos A
1 sinA
1 sinA
cos A 2 secA
cos2 A (1 sinA)(1 sinA)
(1 sinA)(cosA)
cos2 A (1 2sinA sin2 A)
(1 sinA)(cosA)
cos2 A sin2 A 1 2sinA
(1 sinA)(cosA)
2 2sinA
(1 sinA)(cosA)
2(1 sinA)
(1 sinA)(cosA)
2
(cosA)
2secA
2secA
L.S. = R.S.
Using Exact Values to Prove an Identity
5.4.11
Consider sinx
1 cos x
1 cosx
sinx.
b) Verify that this statement is true for x =
6.
a) Use a graph to verify that the equation is an identity.
c) Use an algebraic approach to prove that the identity is true
in general. State any restrictions.
y 1 cosx
sinxy
sinx
1 cosxa)
sinx
1 cosx
1 cosx
sinx
1
2
1 3
2
b) Verify that this statement is true for x =
6.
sin
6
1 cos
6
1
2
2
2 3
1
2 3
1 cos
6
sin
6
1 3
2
1
2
2 3
2
2
1
2 3
2 3
1
2 3
2 3
2 3
2 3
4 3
2 3
Rationalize the
denominator:
1
2 3
L.S. = R.S.
Using Exact Values to Prove an Identity [cont’d]
5.4.12
Therefore, the identity is
true for the particular
case of x
6.
c) Use an algebraic approach to prove that the identity is true
in general. State any restrictions.
Using Exact Values to Prove an Identity [cont’d]
5.4.13
sinx
1 cosx
1 cosx
sinx
sinx
1 cosx
1 cosx
1 cosx
sinx(1 cosx)
1 cos2
x
sinx(1 cosx)
sin2x
1 cosx
sinx
1 cosx
sinx
L.S. = R.S.
Note the left side of the
equation has the restriction
1 - cos x ≠ 0 or cos x ≠ 1.
Therefore, x ≠ 0 + 2 n,
where n is any integer.
The right side of the
equation has the restriction
sin x ≠ 0. x = 0 and Therefore, x ≠ 0 + 2n
and x ≠ + 2n, where
n is any integer.
Restrictions:
Proving an Equation is an Identity
Consider the equation sin2 A 1
sin2
A sinA 1
1
sinA.
b) Verify that this statement is true for x = 2.4 rad.
a) Use a graph to verify that the equation is an identity.
c) Use an algebraic approach to prove that the identity is true
in general. State any restrictions.
y sin2 A 1
sin2
A sinAy 1
1
sinA
a)
5.4.14
b) Verify that this statement is true for x = 2.4 rad.
Proving an Equation is an Identity [cont’d]
sin2 A 1
sin2
A sinA 1
1
sinA
(s in 2.4)2 1
(s in 2.4)2
sin2.4
= 2.480 466
1
1
sin 2.4
= 2.480 466
Therefore, the equation is true for x = 2.4 rad.
L.S. = R.S.
5.4.15
5.4.16
Proving an Equation is an Identity [cont’d]
sin2 A 1
sin2
A sinA 1
1
sinA
c) Use an algebraic approach to prove that the identity is true
in general. State any restrictions.
(s inA 1)(sinA 1)
sinA(s inA 1)
(sinA1)
sinA
sinA
sinA
1
sinA
1 1
sinA
1 1
sinA
L.S. = R.S.
Note the left side of the
equation has the restriction:
sin2A - sin A ≠ 0
A 0, or A
2
Therefore, A 0 2 n or
A + 2n, or
A
2 2 n, where n is
any integer.
The right side of the
equation has the restriction
sin A ≠ 0, or A ≠ 0.
Therefore, A ≠ 0, + 2 n,
where n is any integer.
sin A(sin A - 1) ≠ 0
sin A ≠ 0 or sin A ≠ 1
Applications of Trigonometry This field of mathematics can be applied in
astronomy,navigation, music theory, acoustics, optics, analysis of financial markets, electronics, probability theory, statistics, biology, medical imaging (CAT scans and ultrasound), pharmacy, chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography and in many physical sciences.
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Trigonometry is a branch of Mathematics with several important and useful
applications. Hence it attracts more and more research with several theories
published year after year
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Conclusion