Trigonometry Flashcards
If a and b are the lengths of the legs of a right triangle, and c is the hypotenuse, then…
a2 + b2 = c2
If positive numbers a, b, and c satisfy a2 + b2 = c2, then…
There exists a triangle with sides a, b, and c and this triangle has a right angle opposite the side of c.
If angle C of ΔABC is obtuse, then…
c2 > a2 + b2
The converse applies.
If angle C of ΔABC is acute, then…
c2 < a2 + b2
The converse applies.
sin Θ = ?
sin Θ = opp/hyp
What is the law of sines?
a / sin a = b / sin b = c / sin c
or
sin a / a = sin b / b = sin c / c
cos Θ = ?
cos Θ = adj/hyp
If a + b = 90º, then…
sin a = cos b
and
sin b = cos a
If a + b = 90º, then why is sin a = cos(90 - a)?
b = 90 - a, and since sin a = cos b, we can subsitute b for 90 - a.
sin2(x) + cos2(x) = ?
sin2(x) + cos2(x) = 1
sin2(x) + cos2(x) = 1, why?
Prove that sin a + cos a is always less than 2.
sin a or cos a can never be above 1, and geometrically both of them can never be 1 in the same triangle.
Prove sin a + cos a >= 1.
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.
tan Θ = ?
tan Θ = opp/adj
cot Θ = ?
cot Θ = adj/opp
sec Θ = ?
sec Θ = hyp/adj
csc Θ = ?
csc Θ = hyp/opp
What affects the trigonometric ratios of acute angles?
Only the angle itself, as since there are two congruent angles in each triangle, they will be similar. Thus, they have the same ratios.
If sin a = a, find the lengths of the legs and the hypotenuse.
Assuming a is the smallest angle.
The shorter leg is a.
The longer leg is sqrt(1 - a).
The hypotenuse is 1.
tan a = ?
tan a = cot b
sec a = ?
sec a = csc b
sin a / cos a = tan a, why?
(a/c) / (b/c) = a/b
tan a = a/b,
thus sin a / cos a = tan a
sin a / cos a = ?
sin a / cos a = tan a
cos a / sin a = ?
cot a = cos a / sin a,
same reasoning for tan a = sin a / cos a
(tan a)(cot a) = ?
(tan a)(cot a) = 1
1/tan a = ?
cot a = 1/tan a
tan2a + 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
(cot a)(sin a) = ?
(cot a)(sin a) = cos a
(tan a)/(sin a) = 1/(cos a), why?
(a/b)/(a/c)
ac/ab
c/b = 1/cos a
cos2a - sin2 a = 2 cos2 a - 1, why?
cos2 a - (1 - cos2 a), sin2 a = 1 - cos2 a
2 cos2 a - 1
1 + cot2 a = ?
1/sin2 a
(1-cos a)/(1 + cos a) = (sin a / 1 + cos a)2, why?
1 - cos2 a / (1 + cos a)2, multiply by 1+cos a
sin2 a / (1 + cos a)2 , sin2 a + cos2a = 1
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
If a is the angle subtended by a chord PB at a point on the circle of radius r, then…
sin a = PB/2r
The sine of an obtuse angle is equal to the sine of its…
supplement
1/2 * a * b sin σ = ?
S, the area of a triangle
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.
b2 = a2 + c2 - 2ac cos B, why?
The cosine of an obtuse angle is the cosine of its supplement is:
multiplied by -1
cos(360 + a) = ?
cos a
sin(360 + a) = ?
sin a
A function is odd when…
f(-x) = -f(x)
A function is even when…
f(-x) = f(x)
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.
