Trigonometry

Advanced sin, cos, tan functions

For △DEF shown, which of the following equations, when solved for x, gives the length of DE in inches? (∠E = 45°, ∠F = 73°, and side FE = 13 in.) (Note: sin D / d = sin E / e = sin F / f)

2026 April(R4A) Math Question 31

The circle shown in the standard (x,y) coordinate plane is centered at the origin and passes through N(1,0). Suppose the circle passes through point P in Quadrant I such that the measure of ∠NOP is π/3 radians. What is the y-coordinate of P?

2026 April(R4A) Math Question 34

43. If 90° < θ < 180° and sin θ = 8/17 , then cos θ = ?

Sine, Cosine and Tangent in Four Quadrants

2020 December(D03) Math Question 43

44. Li is standing at point L on the north side of the small canyon shown in the figure below. As measured by line of sight, Li is 2 miles from an observation tower at 7', and she is 3 miles from a scenic overlook at S. Li, the observation tower, and the scenic overlook are all at the same elevation. The measure of LS is 28°. Which of the following equations, when solved, gives the measure of LT ?

2013 June(71C) Math Question 44

46. For all real values of x, which of the following equations is true?

2018 June(A11) Math Question 46

2020 June(C02) Math Question 46

2011 April(67F) Math Question 48

53. A forest fire is contained within a triangular region,
 which is shown below. The supervising firefighter plans
to fight the fire by positioning a firefighter about every
4 meters along the perimeter of the triangle. Among the
following, which expression best estimates the planned
number of firefighters along the perimeter? ·
(Note: The law of sines states that in every triangle, the
3 ratios of length of a side to the sine of the angle
opposite that side are equal.)

2019 June(B02) Math Question 53

2019 June(72F) Math Question 53

2010 April(68G) Math Question 53

2009 December(68A) Math Question 53

53. In the figure below, the distances between 2 pairs of cities are shown, as well as the angle formed at Ewing, which has a measure of 127°. Which of the following values is closest to the distance, in miles, from Deerborn to Fergus?

2021 June(D06) Math Question 53

54. One of the following graphs in the standard (x.y) coordinate plane k the graph of v = sin2x + cos2x over the domain —It-- < x < . Which one?

2017 April(74F) Math Question 54

2013 December(71E) Math Question 54

2010 April(68G) Math Question 57

2009 June(67C) Math Question 57

58. The lengths of the sides of the triangle shown below are given in meters. Which of the following equations gives the degree measure 0 ? (Note: For any triangle, c^2 = a^2 + b^2 — 2ab cos C, where a, b, and c are the lengths of the sides opposite angles with measures A, B, and C, respectively.)

2014 April(71H) Math Question 58

2011 June(69F) Math Question 58

58. The sides of an acute triangle measure 14 cm, 18 cm, and 20 cm, respectively. Which of the following equations, when solved for 0, gives the measure of the smallest angle of the triangle?

Advanced sin, cos, tan functionsTrigonometryACT math questions

2026 April(R4A) Math Question 31

2026 April(R4A) Math Question 34