Trigonometry

Question 1

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

Answer 1

a² + b² = c²

Question 2

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

Answer 2

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

Question 3

If angle C of ΔABC is obtuse, then…

Answer 3

c² > a² + b²

The converse applies.

Question 4

If angle C of ΔABC is acute, then…

Answer 4

c² < a² + b²

The converse applies.

Question 5

sin Θ = ?

Answer 5

sin Θ = opp/hyp

Question 6

What is the law of sines?

Answer 6

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

or

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

Question 7

cos Θ = ?

Answer 7

cos Θ = adj/hyp

Question 8

If a + b = 90º, then…

Answer 8

sin a = cos b

and

sin b = cos a

Question 9

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

Answer 9

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

Question 10

sin²(x) + cos²(x) = ?

Answer 10

sin²(x) + cos²(x) = 1

Question 11

sin²(x) + cos²(x) = 1, why?

Answer 11

Question 12

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

Answer 12

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

Question 13

Prove sin a + cos a >= 1.

Answer 13

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.

Question 14

tan Θ = ?

Answer 14

tan Θ = opp/adj

Question 15

cot Θ = ?

Answer 15

cot Θ = adj/opp

Question 16

sec Θ = ?

Answer 16

sec Θ = hyp/adj

Question 17

csc Θ = ?

Answer 17

csc Θ = hyp/opp

Question 18

What affects the trigonometric ratios of acute angles?

Answer 18

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

Question 19

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

Answer 19

Assuming a is the smallest angle.

The shorter leg is a.

The longer leg is sqrt(1 - a).

The hypotenuse is 1.

Question 20

tan a = ?

Answer 20

tan a = cot b

Question 21

sec a = ?

Answer 21

sec a = csc b

Question 22

sin a / cos a = tan a, why?

Answer 22

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

tan a = a/b,

thus sin a / cos a = tan a

Question 23

sin a / cos a = ?

Answer 23

sin a / cos a = tan a

Question 24

cos a / sin a = ?

Answer 24

cot a = cos a / sin a,

same reasoning for tan a = sin a / cos a

Question 25

(tan a)(cot a) = ?

Answer 25

(tan a)(cot a) = 1

Question 26

1/tan a = ?

Answer 26

cot a = 1/tan a

Question 27

tan²a + 1 = 1/cos² a, why

Answer 27

(sin² a)/(cos² a) + (cos² a)/(cos² a)

1/(cos² a), as sin² a + cos² a = 1

or it is equal to

sec² a

Question 28

(cot a)(sin a) = ?

Answer 28

(cot a)(sin a) = cos a

Question 29

(tan a)/(sin a) = 1/(cos a), why?

Answer 29

(a/b)/(a/c)

ac/ab

c/b = 1/cos a

Question 30

cos²a - sin² a = 2 cos² a - 1, why?

Answer 30

cos² a - (1 - cos² a), sin² a = 1 - cos² a

2 cos² a - 1

Question 31

1 + cot² a = ?

Answer 31

1/sin² a

Question 32

(1-cos a)/(1 + cos a) = (sin a / 1 + cos a)², why?

Answer 32

1 - cos² a / (1 + cos a)², multiply by 1+cos a

sin² a / (1 + cos a)² , sin² a + cos²a = 1

Question 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

Answer 33

Question 34

If a is the angle subtended by a chord PB at a point on the circle of radius r, then…

Answer 34

sin a = PB/2r

Question 35

The sine of an obtuse angle is equal to the sine of its…

Answer 35

supplement

Question 36

1/2 * a * b sin σ = ?

Answer 36

S, the area of a triangle

Question 37

2S/bc = ?, where S is the area of a triangle and b and c are the sides of the triangle.

Answer 37

2S/bc = sin a, where angle a is between b and c.

Question 38

b² = a² + c² - 2ac cos B, why?

Answer 38

Question 39

The cosine of an obtuse angle is the cosine of its supplement is:

Answer 39

multiplied by -1

Question 40

cos(360 + a) = ?

Answer 40

cos a

Question 41

sin(360 + a) = ?

Answer 41

sin a

Question 42

A function is odd when…

Answer 42

f(-x) = -f(x)

Question 43

A function is even when…

Answer 43

f(-x) = f(x)

Question 44

What is a radian measure of an angle?

Answer 44

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.