Trigonometry

2009 April(66F) Math Question 1

5. What is the length,. in. inches, of the hypotenuse of a
right triangle with a leg that is 9 inches long and a leg
that is 2 inches long?

2019 June(B02) Math Question 5

6. The lengths of the legs of a right triangle are 4 miles and 5 miles, respectively. Which of the following lengths, in miles, is closest to that of the hypotenuse of the right triangle?

2015 April(73G) Math Question 6

2012 December(71A) Math Question 6

What is the length, in inches, of the hypotenuse of a right triangle with a leg that is 8 inches long and a leg that is 3 inches long?

2024 September(H11) Math Question 6

2019 April(B04) Math Question 8

2009 December(68A) Math Question 9

9. Right triangle ABC and its side lengths given in inches are shown below. What is sinB?

2021 June(D06) Math Question 9

10. One leg of a right triangle is 4 meters long and the other leg is 8 meters long. The hypotenuse of the triangle is how many meters long?

2014 December(72G) Math Question 10

12. The lengths of the 3 sides of right triangle ABC shown below are given in meters. What is sin,A ?

Trigonometry for Right Triangles

2021 December(E23) Math Question 12

2020 July(C01) Math Question 13

A 5-foot awning extends 2 feet horizontally over the entrance to the vertical building shown below. Which of the following expressions has a value equal to the measure of the angle θ between the building and the awning?

35 February(Z14) Math Question 13

13. In the figure shown below, C is on the segment with endpoints A and D. The distance between A and B is 2,000 km, between A and C is 1,600 km, between A and D is 2,500 km, and between B and C is 1,200 km. What is the distance, in kilometers, between B and D ?

Pythagorean Theorem and Special Right Triangles

2021 December(E23) Math Question 13

2019 April(Z15) Math Question 14

14. Given that sin2x = —4 what is cos2x ?

2016 December(74H) Math Question 14

15. Josiah stands on level ground 15 ft from the base of a cliff. The angle of elevation from where Josiah is standing to the top of the cliff is 50°, as shown below. Which of the following values is closest to the height, in feet, of the cliff? (Note: sin50° ≈ 0.8; cos50° ≈ 0.6; tan50° ≈ 1.2)

2021 April(D05) Math Question 15

17. For an angle with measure a in a right triangle, sin a = 113 15 and tan a = 112/15. What is the value of cos a ?

2014 December(72G) Math Question 17

2020 June(C02) Math Question 17

17. Juanita walked from her home to the bakery, first walking 0.3 miles

2020 December(D03) Math Question 17

17. In right triangle nABC shown below, the given lengths are in millimeters. What is sin,A ?

2021 April(Z04) Math Question 17

The function y = 2 sin(6πx) is graphed in the standard (x,y) coordinate plane for x radians in the interval 0 ≤ x ≤ 1. What are the period and amplitude of the function?

2025 October(J08) Math Question 17

19. In the figure below, A is on BE and C is on BD. What is the measure of ZABC ?

2017 December(A10) Math Question 19

19. Which of the following is the sine of the smallest angle in a right triangle with side lengths 7, 24, and 25 inches, respectively?

2018 April(A09) Math Question 19

2010 April(68G) Math Question 19

19. The lengths of the 2 legs of right triangle AABC shown below are given in inches. The midpoint of AB is how many inches from A ?

2017 April(74F) Math Question 19

A 12-foot taut wire has one end attached to the ground and the other end attached to a vertical pole, as shown below. The point of contact of the wire and the pole is 8 feet above the ground. What angle does the wire make with the level ground?

2024 June(G20) Math Question 19

2019 April(B04) Math Question 20

2009 April(66F) Math Question 20

20. The graph • of y = 3 — 5 sin(x — n) is shown in the standard (x ,y) coordinate plane below. What is the range of y ?

2017 June(74C) Math Question 20

2020 June(C02) Math Question 20

20. In DEF, the length of DE is VT:31 inches, and the

2017 April(74F) Math Question 20

20. Carpenters use the term pitch to describe the slope of a roof. For example, a roof with a pitch of 17-1 means the roof has 1 foot of vertical rise for every 4 feet of horizontal distance. The figure below shows a 13-foot-long roof with 5 feet of vertical rise and x feet of horizontal distance. What is the pitch of this roof?

2016 April(73E) Math Question 20

A surveyor is standing on ground that is level with the base of the cell phone tower shown below. From the point on the ground where the surveyor is standing, the angle of elevation to the top of the tower is 40°. The distance from the surveyor to the tower is 200 feet. Which of the following values is closest to the height, in feet, of the tower? (Note: sin 40° = 0.64, cos 40° = 0.77, and tan 40° = 0.84)

2023 April(F11) Math Question 20

2013 December(71E) Math Question 21

2011 June(69F) Math Question 21

21. The side lengths of right triangle ALMN are given in centimeters in the figure below. What is tan N ?

2014 April(71H) Math Question 21

2018 December(B05) Math Question 22

2009 June(67C) Math Question 22

2019 December(C03) Math Question 24

2012 April(70G) Math Question 24

2009 December(68A) Math Question 24

2019 June(72F) Math Question 24

2020 June(C02) Math Question 24

Given that sin A = 24/25, which of the following values could tan A equal?

2023 April(F11) Math Question 24

25. Right triangle △RST has its right angle at vertex S. The length of ST is 6.0 feet and the length of RS is 2.5 feet. Which of the following values is closest to the length, in feet, of RT ?

2014 June(72C) Math Question 25

2019 April(B04) Math Question 25

2013 December(71E) Math Question 25

2012 April(70G) Math Question 25

2024 December(H31) Math Question 25

26. In 'the figure shown below, a ladder 15 feet long formi
an angle of 56° with the level ground as it leans against
the vertical side of a building. The distance along the
building, fa feet, between the ground and the top of the
ladder is equal to which of the following expressions?

2019 June(B02) Math Question 26

2019 December(C03) Math Question 26

2012 December(71A) Math Question 26

2012 June(70C) Math Question 26

Jabari is walking along a straight sidewalk when he sees a tree directly ahead of him. He estimates that the angle of elevation from his feet to the top of the tree is 25°. As he continues walking, he finds that he was about 45 feet from the base of the tree when he estimated the angle of elevation. Based on Jabari's estimates, which of the following expressions represents the height, in feet, of the tree?

2025 October(J08) Math Question 26

For the triangle shown below, what is tan ∠A?

2023 September(G01) Math Question 26

2009 June(67C) Math Question 27

For an angle with measure α in a right triangle, sin α = 144/145 and tan α = 144/17. What is the value of cos α?

2024 December(H31) Math Question 27

2012 June(70C) Math Question 28

2011 June(69F) Math Question 28

29. Nestor will bury one end of a cable 3 feet from the base of an antenna and attach the other end of the cable at a point on the antenna 8 feet above the ground, as shown below. When taut, the length of the exposed cable will be Ni. feet. Which of the following expressions represents the measure of the angle the taut cable will make with the level ground?

2015 April(73G) Math Question 29

29. At Central High School, 4 out of every 10 students ride the bus to and from school, and 3 out of every 8 who ride the bus are freshmen. If there are 2,500 students at Central, how many of the students are freshmen who ride the bus?

2017 December(A10) Math Question 29

2010 April(68G) Math Question 29

29. Paula is planning a course for a bike race. The course is in the shape of a right triangle, as shown below. Participants will begin at A, ride directly to B, then directly to C, and directly back to A. Paula wants to put a rest stop at the halfway point on the course. How many miles past B will the rest stop be?

2018 April(A09) Math Question 29

29. Janelle is loading a truck by using a ramp, as shown below. The ramp is 8 feet long, and the end of the ramp that is resting on the truck is 2.5 feet above the level ground. Which of the following expressions gives the angle of inclination of the ramp?

2014 June(72C) Math Question 29

29. For right triangle AABC shown below, which of the following expressions has a value that is equal to cos A ?.

2013 June(71C) Math Question 29

30. The base of an escalator in a store is 26 meters long and has a vertical lift of 10 meters as shown below. Which of the following expressions is closest to the angle of inclination between the base of the escalator and the horizontal floor?

2016 April(73E) Math Question 30

30. For an angle with measure a in a right triangle, sin a = 40/41 and tan a = 40/9. What is the value of cos a ?

2015 June(73C) Math Question 30

Let θ be an acute angle of a right triangle. Given sin θ = a/b and tan θ = a/c, cos θ = ?

2024 June(G20) Math Question 30

30. In right triangle nABC shown below, sin,C = and the length of AB is 14 inches. What is the length, in inches, of AC ?

The law of sines & The law of cosines

2020 December(D03) Math Question 30

2019 June(72F) Math Question 31

2012 June(70C) Math Question 31

2012 April(70G) Math Question 31

2009 June(67C) Math Question 31

31. In ABC shown below, sin C = 4/5 and the length of AB is 10 inches. What is the length, in inches, of AC ?

2015 December(72E) Math Question 31

32. In the right triangle AABC shown below, the length of

2018 April(A09) Math Question 32

In the figure below, a highway rest area (at D) and radar stations (at A and B) lie on a level east-west line; A is 9,000 feet due west of D. An airplane (at C) is shown directly above the rest area, flying due west at a constant speed of 300 feet per second and at a constant altitude of 12,000 feet. The airplane is located at,a straight-line distance of 15,000 feet from the radar station at A and 13,000 feet from the radar station at B.

2017 December(A10) Math Question 32

32. Which of the following ratios is equal to cos B ? (Note: FG denotes the length of FG .)

2016 April(73E) Math Question 32

32. The radius of the base of the right circular cone shown below is 5 inches, and the height of the cone is 7 inches. Solving which of the following equations gives the measure, 0, of the angle formed by a slant height of the cone and a radius?

2014 June(72C) Math Question 32

32. Olivia, Ashton, and Jane are standing on a soccer field
such that Olivia is 20 meters due west of Ashton and
Jane is 40 meters due north of Ashton. Their positions
are at the vertices of a triangle. Which of the following
expressions gives the degree measure of the angle of
the triangle at the vertex where Olivia is standing?

2019 June(B02) Math Question 32

33. The perimeter of the pyramid's base was 920 meters when construction was completed. At that time, which of the following values is closest to the length, in meters, of each diagonal of the base?

2016 December(74H) Math Question 33

2013 April(71G) Math Question 33

2012 December(71A) Math Question 33

2010 December(69A) Math Question 33

2020 July(C01) Math Question 33

33. Suppose ABCD is placed in the standard (x,y) coordinate plane such that C is at (0,0), B is at (25,0), and A and D have positive x- and y-coordinates. What is the x-coordinate of D ?

2016 April(73E) Math Question 33

For the right triangle below, tan θ = 5/√39. What is sin θ?

2024 April(Z13) Math Question 33

34. For an angle with measure a in a right triangle, sin a = 180/181 and tan a = 180/19. What is the value of cos a?

2015 April(73G) Math Question 34

34. Which of the following expressions is equal to the length, in meters. of DC ?

2016 December(74H) Math Question 34

2019 June(72F) Math Question 34

2010 December(69A) Math Question 34

34. What is the length, in meters, of JD?

2015 June(73C) Math Question 34

2009 December(68A) Math Question 35

2012 December(71A) Math Question 36

2013 April(71G) Math Question 36

2012 April(70G) Math Question 37

2009 April(66F) Math Question 37

37. Rectangle JKLM is shown below. The given side lengths are in feet. What is the cosine of ∠KJL ?

2021 April(D05) Math Question 37

37. Two motorcycles, starting at the same point at the same time, travel away from each other at a 90° angle. One travels at 40 miles per hour and the other at 60 miles per hour. If they continue traveling at these constant rates, after about how many hours will they be 200 miles apart?

Speed, Distance, and Time

2020 December(D03) Math Question 37

37. In nDEF, shown in the figure below, EG ___ is an altitude, ∠DEF is a right angle, EF = 20 centimeters, and the measure of ∠EDF is 30°. What is EG, in centimeters?

2021 December(E23) Math Question 37

38. The figure below shows a 10-foot ladder leaning against a vertical wall. The base of the ladder makes a 65° angle with the level ground. Which of the following expressions gives the distance, in feet, between the base of the ladder and the wall?

TrigonometryThere are 208 ACT math questions under Trigonometry

Sub-topics

2009 April(66F) Math Question 1

2019 June(B02) Math Question 5