Download Trigonometric Functions of Right Triangles — 6.2 θ

Math 4 Pre-Calculus
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Trigonometric Functions of Right Triangles — 6.2
Definitions
A triangle is a right triangle if one of its angles is a right angle. See pictures below.
The sine of θ , denoted s i n θ , is the ratio of the side opposite θ to the hypotenuse.
The cosine of θ , denoted c o s θ , is the ratio of the side adjacent θ to the hypotenuse.
The tangent of θ , denoted t a n θ , is the ratio of the side opposite θ to the side adjacent θ .
In summary:
sinθ =
opp
h yp
co sθ =
adj
h yp
You can remember this by: SOH CAH TOA
1.
Find sine, cosine and tangent for the angle θ .
2.
If θ is an acute angle and s i n θ =
3
, find c o s θ and t a n θ .
4
tanθ =
opp
adj
Reciprocal Trigonometric Functions
The cosecant of θ , denoted c s c θ , is the reciprocal of the sine θ or the ratio of the hypotenuse to the side opposite θ .
The secant of θ , denoted s e c θ , is the reciprocal of the cosine θ or the ratio of the hypotenuse to the side adjacent θ .
The cotangent of θ , denoted c o t θ , is the reciprocal of the tangent θ or the ratio of the side adjacent θ to the side opposite θ .
In summary:
cscθ =
1
h yp
=
sinθ
opp
3.
Find all six trigonometric functions for the angle θ .
s e cθ =
1
hyp
=
cosθ
adj
cotθ =
1
adj
=
tanθ
opp
θ
y
x
4.
Label all of the angles of the two triangles both in degrees and radians, and then label the lengths of each side.
Isosceles Right Triangle
30-60-90 Triangle
Use the isosceles right triangle and the 30-60-90 triangle above to fill in the chart below.
θ (radians) θ (degrees)
30°
π
4
60°
sinθ
cosθ
tanθ
cotθ
s e cθ
cscθ
5.
Find exact values of x and y .
6.
Distance to Mt. Fuji The peak of Mt. Fuji in Japan is approximately 12,400 feet high. A trigonometry student,
several miles away notes that the angle between level ground and the peak is 30°. Estimate the distance from the
student to the point on level ground directly beneath the peak.
7.
Approximate to four decimal places, when appropriate.
a)
sin 75°
b)
cot
π
5
Fundamental Trigonometric Identities
Reciprocal Identities:
1
1
cscθ =
secθ =
sinθ
cosθ
Tangent and Cotangent Identities:
sinθ
tanθ =
co sθ
Pythagorean Identities:
s i n 2 θ + c o s2 θ = 1
sinθ
.
co sθ
8.
Show t a n θ =
9.
Show s i n 2 θ + c o s 2 θ = 1
cotθ =
cotθ =
1
tanθ
cosθ
sinθ
1 + t a n 2 θ = s e c2 θ
1 + c o t 2 θ = c s c2 θ
10.
Use the Pythagorean identities to write the expression as an integer.
( 2 ) − 3cot (β 2 )
3 c s c2 β
11.
2
Simplify the expression.
(
c s cθ + 1
1
s i n2 θ
) + cscθ
12.
Use the fundamental identities to write the first expression in terms of the second, for any acute angle θ .
cosθ , sinθ
13.
Verify the identity by transforming the left-hand side into the right-hand side.
tanθ cotθ = 1
14.
Verify the identity by transforming the left-hand side into the right-hand side.
c o s 2 ( 2θ ) − s i n 2 ( 2θ ) = 2 c o s 2 ( 2θ ) − 1
15.
Verify the identity by transforming the left-hand side into the right-hand side.
co t θ + t a n θ = c s cθ s e cθ