Trigonometry Flashcards

1
Q

If a and b are the lengths of the legs of a right triangle, and c is the hypotenuse, then…

A

a2 + b2 = c2

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2
Q

If positive numbers a, b, and c satisfy a2 + b2 = c2, then…

A

There exists a triangle with sides a, b, and c and this triangle has a right angle opposite the side of c.

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3
Q

If angle C of ΔABC is obtuse, then…

A

c2 > a2 + b2

The converse applies.

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4
Q

If angle C of ΔABC is acute, then…

A

c2 < a2 + b2

The converse applies.

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5
Q

sin Θ = ?

A

sin Θ = opp/hyp

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6
Q

What is the law of sines?

A

a / sin a = b / sin b = c / sin c

or

sin a / a = sin b / b = sin c / c

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7
Q

cos Θ = ?

A

cos Θ = adj/hyp

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8
Q

If a + b = 90º, then…

A

sin a = cos b

and

sin b = cos a

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9
Q

If a + b = 90º, then why is sin a = cos(90 - a)?

A

b = 90 - a, and since sin a = cos b, we can subsitute b for 90 - a.

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10
Q

sin2(x) + cos2(x) = ?

A

sin2(x) + cos2(x) = 1

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11
Q

sin2(x) + cos2(x) = 1, why?

A
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12
Q

Prove that sin a + cos a is always less than 2.

A

sin a or cos a can never be above 1, and geometrically both of them can never be 1 in the same triangle.

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13
Q

Prove sin a + cos a >= 1.

A

sin a + cos a >= 1, square both sides

sin2 a + cos2 a + 2 cos a sin a >= 1

1 + 2 cos a sin a >= 1

Therefore, equality proven.

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14
Q

tan Θ = ?

A

tan Θ = opp/adj

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15
Q

cot Θ = ?

A

cot Θ = adj/opp

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16
Q

sec Θ = ?

A

sec Θ = hyp/adj

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17
Q

csc Θ = ?

A

csc Θ = hyp/opp

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18
Q

What affects the trigonometric ratios of acute angles?

A

Only the angle itself, as since there are two congruent angles in each triangle, they will be similar. Thus, they have the same ratios.

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19
Q

If sin a = a, find the lengths of the legs and the hypotenuse.

A

Assuming a is the smallest angle.

The shorter leg is a.

The longer leg is sqrt(1 - a).

The hypotenuse is 1.

20
Q

tan a = ?

A

tan a = cot b

21
Q

sec a = ?

A

sec a = csc b

22
Q

sin a / cos a = tan a, why?

A

(a/c) / (b/c) = a/b

tan a = a/b,

thus sin a / cos a = tan a

23
Q

sin a / cos a = ?

A

sin a / cos a = tan a

24
Q

cos a / sin a = ?

A

cot a = cos a / sin a,

same reasoning for tan a = sin a / cos a

25
(tan a)(cot a) = ?
(tan a)(cot a) = 1
26
1/tan a = ?
cot a = 1/tan a
27
tana + 1 = 1/cos2 a, why
(sin2 a)/(cos2 a) + (cos2 a)/(cos2 a) 1/(cos2 a), as sin2 a + cos2 a = 1 or it is equal to sec2 a
28
(cot a)(sin a) = ?
(cot a)(sin a) = cos a
29
(tan a)/(sin a) = 1/(cos a), why?
(a/b)/(a/c) ac/ab c/b = 1/cos a
30
cosa - sin2 a = 2 cos2 a - 1, why?
cos2 a - (1 - cos2 a), sin2 a = 1 - cos2 a 2 cos2 a - 1
31
1 + cot2 a = ?
1/sin2 a
32
(1-cos a)/(1 + cos a) = (sin a / 1 + cos a)2, why?
1 - cos2 a / (1 + cos a), multiply by 1+cos a sin2 a / (1 + cos a)2 , sin2 a + cosa = 1
33
If a is the angle subtended by a chord PB at a point on the circle of radius r, then sin a = PB/2r, why
34
If a is the angle subtended by a chord PB at a point on the circle of radius r, then...
sin a = PB/2r
35
The sine of an obtuse angle is equal to the sine of its...
supplement
36
1/2 \* a \* b sin σ = ?
S, the area of a triangle
37
2S/bc = ?, where S is the area of a triangle and b and c are the sides of the triangle.
2S/bc = sin a, where angle a is between b and c.
38
b2 = a2 + c2 - 2ac cos B, why?
39
The cosine of an obtuse angle is the cosine of its supplement is:
multiplied by -1
40
cos(360 + a) = ?
cos a
41
sin(360 + a) = ?
sin a
42
A function is odd when...
f(-x) = -f(x)
43
A function is even when...
f(-x) = f(x)
44
What is a radian measure of an angle?
A radian measure of an angle is the ratio of the arc it cuts off to the radius of any circle who center is the vertex.
45