Trigonometric Ratios
Cofunctions
Using Trigonometry to Find a Side
Using Trigonometry to Find an Angle
Using Trigonometry to Find Area
100
In triangle ABC, the measure of ∠A = 90°, AB = 6, AC = 8, and BC = 10.
Which ratio represents the sine of ∠B?
What is 4/5?
Explanation: side AC is the opposite of ∠B and side BC is the hypotenuse.
SOH - sine: opposite/hypotenuse
sine of ∠B is 8/10, simplified is 4/5
100
In scalene triangle ABC shown in the diagram below, m∠C = 90°.
Which equation is always true?
1) sinA = sinB
2) cosA = cosB
3) cosA = sinC
4) sinA = cosB
What is 4) sinA = cosB?
100
In right triangle ABC below, m∠C = 90°, AC = 12, and m∠A = 25°.
1) a = 12/tan25°
2) a = 12tan25°
3) c = 12/tan25°
4) c = 12tan25°
What is 2) a = 12tan25°?
Explanation:
tan25° = a/12
12tan25° = a
a = 12tan25°
100
In the diagram of right triangle ABC shown below, AB = 14 and AC = 9.
What is the measure of ∠A, to the nearest degree?
1) 33
2) 40
3) 50
4) 57
What is 3) 50?
Explanation: CAH - cosine (adjacent/hypotenuse)
cos-1(9/14) = x
x is equivalent to about 50 - answer choice 3)50.
100
In non-right triangle ABC shown below, AC = 5 in, BC = 8 in, and m∠C = 57°.
What is the area of ABC, to the nearest tenth of a square inch?
1) 10.9
2) 16.8
3) 21.8
4) 33.5
What is 2) 16.8?
Explanation: use the formula A(area) = ab/2 (sine(C)) to solve.
8(5)/2 x (sin(57)) = A
The area is equivalent to about 16.8 once rounded - answer choice 2) 16.8.
200
The diagram below shows right triangle UPC.
Which ratio represents the sine of ∠U?
1) 15/8
2) 15/17
3) 8/15
4) 8/17
What is 2) 15/17?
Explanation: SOH - sine (opposite/hypotenuse).
200
In ABC, the complement of ∠B is ∠A. Which statement is always true?
1) tan∠A = tan∠B
2) sin∠A = sin∠B
3) cos∠A = tan∠B
4) sin∠A = cos∠B
What is 4) sin∠A = cos∠B?
200
As shown in the diagram below, the angle of elevation from a point on the ground to the top of the tree is 34°.
If the point is 20 feet from the base of the tree, what is the height of the tree, to the nearest tenth of a foot?
1) 29.7
2) 16.6
3) 13.5
4) 11.2
What is 3) 13.5?
Explanation: TOA - tangent (opposite/adjacent) (in this case is height of the tree - x, over 20 ft).
tan(34) = x/20
20(tan(34)) = x
x is equivalent to about 13.5 - answer choice 3) 13.5.
200
In the diagram of triangle RST below, m∠T = 90°, RS = 65, and ST = 60.
What is the measure of ∠S, to the nearest degree?
1) 23º
2) 43º
3) 47º
4) 67º
What is 1) 23º?
Explanation: CAH - cosine (adjacent/hypotenuse).
cos-1(60/65) = x
x is equivalent to about 23 - answer choice 1) 23º.
200
In triangle RST, m∠S = 135, r = 27, and t = 19. What is the area of triangle RST to the nearest tenth of a square unit?
1) 90.7
2) 181.4
3) 256.5
4) 362.7
What is 2) 181.4?
Explanation: use the formula A(area) = ab/2(sin(C))
A = 27(19)/2 x (sin(135))
A is equivalent to about 181.4 - answer choice 2) 181.4.
300
In right triangle ABC shown below, what is the value of cosA?
1) 12/20
2) 16/20
3) 20/12
4) 20/16
What is 2)16/20?
Explanation: CAH - cosine (adjacent/hypotenuse).
300
Which expression is always equivalent to sinx when 0° < x < 90°?
1) cos(90° − x)
2) cos(45° − x)
3) cos(2x)
4) cos x
What is 1) cos(90° − x)?
300
Yolanda is making a springboard to use for gymnastics. She has 8-inch-tall springs and wants to form a 16.5° angle with the base, as modeled in the diagram below.
To the nearest tenth of an inch, what will be the length of the springboard, x?
1) 2.3
2) 8.3
3) 27.0
4) 28.2
What is 4) 28.2?
Explanation: SOH - sine (opposite/hypotenuse) (in this case is 8 in over x).
sin(16.5) = 8/x
x(sin(16.5)) = 8
x = 8/sin(16.5)
x is equivalent to about 28.2 - answer choice 4) 28.2.
300
Many roofs are slanted to prevent the buildup of snow. As modeled below, the length of a roof is 5.5 meters and it rises to a height of 2.5 meters.
The angle of elevation of the roof, to the nearest degree, is
1) 24º
2) 25º
3) 27º
4) 28º
What is 3) 27º?
Explanation: the angle of elevation in this case leads to the use of CAH - cosine (adjacent/hypotenuse).
cos-1(2.5/5.5) = x
x is equivalent to about 27 - answer choice 3) 27º.
300
In parallelogram BFLO, OL = 3.8, LF = 7.4, and m∠O = 126. If diagonal BL is drawn, what is the area of BLF?
1) 11.4
2) 14.1
3) 22.7
4) 28.1
What is 1) 11.4?
Explanation: find the angle supplementary to 126, which is 54 - the angle used to plug into the equation A(area) = ab/2(sin(C)).
A = 7.4(3.8)/2 x (sin(54))
A is equivalent to about 11.4 - answer choice 1) 11.4.
400
In right triangle LMN shown below, m∠M = 90°, MN = 12, and LM = 16.
The ratio of cosN is
1) 12/20
2) 16/20
3) 12/16
4) 16/12
What is 1) 12/20?
Explanation: CAH - cosine (adjacent/hypotenuse).
400
If cos(2x-1)° = sin(3x+6)°, then the value of x is
1) 7
2) 17
3) 35
4) 71
What is 2) 17?
Explanation: make the equation (2x-1)+(3x+6)=90 and solve.
(2x-1)+(3x+6)=90
5x+5=90
5x=85
x=17
400
Triangle ABC has a right angle at C. If AC = 7.7 and m∠B = 24°, what is AB, to the nearest tenth?
1) 18.9
2) 17.3
3) 8.4
4) 3.1
What is 1) 18.9?
Explanation: SOH - sine (opposite/hypotenuse) (in this case is 7.7 over the hypotenuse - x).
sin(24) = 7.7/x
x(sin(24)) = 7.7
x = 7.7/sin(24)
x is equivalent to about 18.9 - answer choice 1) 18.9.
400
In right triangle ABC, hypotenuse AB has a length of 26 cm, and side BC has a length of 17.6 cm. What is the measure of angle B, to the nearest degree?
1) 48°
2) 47°
3) 43°
4) 34°
What is 2) 47°?
Explanation: CAH - cosine (adjacent/hypotenuse)
cos-1(17.6/26) = x
x is equivalent to about 47 - answer choice 2) 47°.
400
Find, to the nearest tenth, the area of triangle ABC if a = 6, b = 10, and m∠C = 18.
What is 9.3?
Explanation: use the formula A(area) = ab/2(sin(C)).
A = 6(10)/2 x (sin(18))
A is equivalent to about 9.3.
500
In the accompanying diagram of right triangle ABC, AB = 8, BC = 15, AC = 17, and m∠ABC = 90.
What is tan∠C?
1) 8/15
2) 17/15
3) 8/17
4) 15/17
What is 1) 8/15?
Explanation: TOA - tangent (opposite/adjacent).
500
Which value of x satisfies the equation sin(3x+5)° = cos(4x+1)°?
1) 30
2) 24
3) 12
4) 4
What is 3)12?
Explanation: make the equation (3x+5)+(4x+1)=90 and solve.
(3x+5)+(4x+1)=90
7x+6=90
7x=84
x=12
500
The angle of elevation from a point 25 feet from the base of a tree on level ground to the top of the tree is 30°. Which equation can be used to find the height of the tree?
1) tan30° = x/25
2) sin30° = x/25
3) cos 30° = x/25
4) 302 + 252 = x2
What is 1) tan30° = x/25?
Explanation: TOA - tangent (opposite/adjacent) (in this case is the height - x, over 25 - the distance from the base of the tree to a point on ground level). This results in the answer choice 1) tan30° = x/25 once everything is plugged in.
500
A man who is 5 feet 9 inches tall casts a shadow of 8 feet 6 inches. Assuming that the man is standing perpendicular to the ground, what is the angle of elevation from the end of the shadow to the top of the man’s head, to the nearest tenth of a degree?
1) 34.1
2) 34.5
3) 42.6
4) 55.9
What is 1) 34.1?
Explanation: convert every value into all inches, leading to 69 in and 102 in.
TOA - tangent (opposite/adjacent)
tan-1(69/102) = x
x is equivalent to about 34.1 - answer choice 1) 34.1.
500
In ABC, a = 12, b = 20.5, and m∠C = 73. Find the area of ABC, to the nearest tenth.
What is 117.6?
Explanation: use the formula A(area) = ab/2(sin(C)).
A = 12(20.5)/2 x (sin(73))
A is equivalent to about 117.6.