Download AN A-Z OF TRIGONOMETRY The unit circle is the curve in the plane

AN A-Z OF TRIGONOMETRY
ADAM PIGGOTT
The unit circle is the curve in the plane described by the equation
x + y 2 = 1. It is a familiar and comfortable shape in the plane. It is
also our tool for describing angles. We describe an angle by describing
the length of the corresponding arc on the unit circle. When we do
this, we have described the angle in radians.
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Question set 1.
(A) What is π?
(B) What is the circumference of the unit circle?
(C) How many radians make a complete rotation?
(D) What does an angle of 30◦ look like?
(E) What is 30◦ in radians?
(F) How do we convert between degrees and radians? And back?
Definition 1 (The cosine function). We define the cosine, usually abbreviated to cos, function by the following rule: for each real number
θ, cos(θ) is the x-coordinate of the point reached by traveling θ units
counter-clockwise around the unit circle from the point (1, 0).
Question set 2.
(G) What is the domain of cos(θ)?
(H) What is the range of cos(θ)?
(I) Use the definition to compute cos(0), cos( π4 ), cos( π2 ), cos( −π
),
4
cos( 5π
).
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(J) Describe how cos(θ) is computed when θ < 0.
(K) Explain why cos(θ) is periodic, and why the period is 2π.
(L) Use the definition to sketch the graph w = cos(θ) (the horizontal
axis, which represents the input to the function, should be labeled
θ, and the vertical axis, which represents the output from the
function, should be labeled w).
Definition 2 (The sine function). We define the sine, usually abbreviated to sin, function by the following rule: for each real number θ,
sin(θ) is the y-coordinate of the point reached by traveling θ units
counter-clockwise around the unit circle from the point (1, 0).
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ADAM PIGGOTT
We define a number of other functions as quotients involving sine
and cosine.
Definition 3 (The other trigonometric ratios). For each real number
θ such that θ ̸= π2 + kπ for any integer k, we define
sin(θ)
1
and sec(θ) =
.
cos(θ)
cos(θ)
(tan is short for tangent, and sec is short for secant).
For each real number θ such that θ ̸= kπ for any integer k, we define
cos(θ)
1
cot(θ) =
and csc(θ) =
.
sin(θ)
sin(θ)
(cot is short for cotangent, and csc is short for cosecant).
tan(θ) =
A note about notation: When writing about trigonometric functions,
we have special notation for exponents. We write sink (θ) instead of
(
)k
sin(θ) .
Question set 3. (M) Solve (find ALL solutions to) the equation
sec(θ) = 2.
(N) Explain how the definitions yield the identity: For each real
number θ, sin(−θ) = − sin(θ).
(O) Explain how the definitions yield the identity: For each real
number θ, sin2 (θ) + cos2 (θ) = 1.
(P) Explain how the definitions yield the identity: For each real
number θ such that θ ̸= π2 + kπ for any integer k, tan2 (θ) + 1 =
sec2 (θ).
(Q) Find a similar identity involving the cosecant and cotangent,
and explain where it comes from.
(R) Determine csc(θ) if cos(θ) = 15 and 0 ≤ θ ≤ π; determine csc(θ)
if cos(θ) = 15 and −π ≤ θ ≤ 0.
(S) Explain why the domain of tan(θ) is as described.
(T) What is the period of tan(θ)?
(U) Sketch the graph of tan(θ).
(V) Sketch the graphs of cot(θ), sec(θ) and csc(θ).
Question set 4. (W) Explain how the definitions can be adjusted
to work with a circle of radius r, rather than the unit circle.
adjacent
(X) Explain the familiar cos(θ) = hypotenuse
“definition” of cos(θ).
Why the quotes?
(Y) Explain how trigonometric functions help us deduce missing information about right-angled triangles.
AN A-Z OF TRIGONOMETRY
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1. Trigonometric Identities
An identity is an equation that holds, no matter the values of the
variables. Here are some identities involving trigonometric functions.
In each case, θ and ϕ are any real numbers for which the expression
makes sense, and k is any integer.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
(20)
(21)
sin(θ + 2kπ)
cos(θ + 2kπ)
tan(θ + kπ)
sin(−θ)
cos(−θ)
tan(−θ)
2
sin (θ) + cos2 (θ)
tan2 (θ) + 1
1 + cot2 (θ)
(π
)
sin
−θ
2
(π
)
cos
−θ
2
(π
)
sin
+θ
2
(π
)
cos
+θ
2
sin(θ + ϕ)
cos(θ + ϕ)
cos(2θ)
cos(2θ)
cos(2θ)
=
=
=
=
=
=
=
=
=
sin(θ)
cos(θ)
tan(θ)
− sin(θ)
cos(θ)
− tan(θ)
1
sec2 (θ)
csc2 (θ)
= cos(θ)
= sin(θ)
= cos(θ)
= − sin(θ)
=
=
=
=
=
sin(θ) cos(ϕ) + cos(θ) sin(ϕ)
cos(θ) cos(ϕ) − sin(θ) sin(ϕ)
cos2 (θ) − sin2 (θ)
2 cos2 (θ) − 1
1 − 2 sin2 (θ)
cos(2θ) − 1
cos2 (θ) =
2
1
−
cos(2θ)
sin2 (θ) =
2
sin(2θ) = 2 sin(θ) cos(θ)
Theorem 1 (The Law of Cosines). If a triangle has sides a, b and c,
and θ is the angle opposite side c, then c2 = a2 + b2 − 2ab cos(θ).
Question set 5.
(Z) Assign each identity (1)-(21) to one of the
following groups: periodicity identities; parity identities; square
identities; complementary angles identities; shift formulas; addition formulas; double-angle formulas.