Question 1 :
Given right triangle ABC above, which of the following is equal to c/b?
A) tanB
B) 1/tanB
C) cosB
D) 1/cosB
Answer :
In all the given answer choices, we have the angle measure B. Considering the interior angle of B in the right triangle ABC above, AC is the opposite side and AB is adjacent side.
Length of the opposite side = AC = b
Length of the adjacent side = AB = c
The ratio between opposite side and adjacent side is "tan".
b/c = tanB
Take reciprocal on both sides.
c/b = 1/tanB
The correct answer choice is (B).
Question 2 :
sinx = cosy
In the equation above, x and y are measured in radians. Which of the following could be x in terms of y?
A) π/2 - y
B) π/2 + y
C) y - π/2
D) π - y
Answer :
Since sinx and cosx are equal, the angles measures x and y are complementary.
x + y = 90° or π/2
x + y = π/2
x = π/2 - y
The correct answer choice is (A).
Question 3 :
What is the value of sin30° - cos60°?
A) 0
B) (1 - √3)/2
C) (√2 - 1)/2
D) (√3 - 1)/2
Answer :
From the trigonometric ratio table, we have
sin30° = 1/2
cos60° = 1/2
sin30° - cos60° = 1/2 - 1/2
= 0
The correct answer choice is (A).
Question 4 :
In the triangle above, AB = BC, what is the value of sinA?
A) 0.28
B) 0.56
C) 0.84
D) 0.96
Answer :
In an isosceles triangle, perpendicular drawn to the unequal side will bisect it.
In the triangle ABC above, since AB = AC, it is an isosceles triangle. So, the perpendicular drawn to the unequal side AC will bisect it.
In the right ΔABD, using Pythagorean Theorem,
AD^{2} + BD^{2} = AB^{2}
7^{2} + BD^{2} = 25^{2}
49 + BD^{2} = 625
BD^{2} = 576
BD^{2} = 24^{2}
BD = 24
In the right ΔABD,
sinA = opposite side/ hypotenuse
= BD/AB
= 24/25
= 0.96
The correct answer choice is (D).
Question 5 :
Given right triangle ABC above, which of the following gives the length of AB in terms of θ?
A) sinθ
B) cosθ
C) tanθ
D) 1/sinθ
Answer :
In the right ΔABC above, considering angle θ at the vertex B,
opposite side = AC
adjacent side = AB
hypotenuse = BC
The length of the hypotenuse is given, that is BC = 1 and we have to find the length of the adjacent side.
The ratio between adjacent side and hypotenuse is "cos".
AB/BC = cosθ
AB/1 = cosθ
AB = cosθ
The correct answer choice is (B).
Question 6 :
In the xy-plane above, a circle with radius 5 has its center at the origin. Point A lies on the circle and has coordinates (m, n). What is n in terms θ?
A) 5sinθ
B) 5cosθ
C) tanθ
D) 5(sinθ + cosθ)
Answer :
In the given figure, draw a perpendicular from point A to x-axis. Let the perpendicular meet x-axis at B.
Considering θ in the right ΔOAB in the figure above,
opposite side = AB = n
adjacent side = OB = m
hypotenuse = OA = 5
The length of the hypotenuse is given, that is OA = 5 and we have to find the length of the opposite side AB, that is n.
The ratio between opposite side and hypotenuse is "sin".
AB/OA = sinθ
n/5 = sinθ
n = 5sinθ
The correct answer choice is (A).
Question 7 :
Right triangle ABC is shown in the xy-plane above. What is the value of tanA?
A) 7/12
B) 3/4
C) 7/9
D) 12/7
Answer :
In the right ΔABC above, considering the angle A,
opposite side = BC
adjacent side = AC
Length of the opposite side :
= BC
= y-coordinate at C - y-coordinate at B
= 4 - (-3)
= 7
Length of the adjacent side :
= AC
= x-coordinate at C - x-coordinate at B
= 7 - (-5)
= 7 + 5
= 12
tanA = opposite side/adjacent side
= BC/AC
= 7/12
The correct answer choice is (A).
Question 8 :
sin24° = cos(3k + 6)°
In the equation above, the angle measures are in degrees. If 0° < k < 90°, what is the value of k?
Answer :
Since sin24° and cos(3k + 6)° are equal, then the angles measures 24° and (3k + 6)° are complementary.
24° + (3k + 6)° = 90°
24 + 3k + 6 = 90
3k + 30 = 90
3k = 60
k = 20
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