Trigonometric Functions of Acute Angles

Concept Check Work each problem.

65. What value of A between 0 degrees and 90 degrees will produce the output shown on the graphing calculator screen?

Image summary: This figure is a screenshot of a digital calculator display. It shows a mathematical expression involving a square root over a denominator, followed by a trigonometric sine function of an angle. The results for both the numerical expression and the sine function are identical, indicating that the value of the sine of angle A is equal to the square root of three divided by two.

66. A student was asked to give the exact value of sine 45 degrees. Using a calculator, he gave the answer 0.7071067812. Explain why the teacher did not give him credit.

67. Find the equation of the line that passes through the origin and makes a 30 superscript circle angle with the x-axis.

68. Find the equation of the line that passes through the origin and makes a 60 superscript circle angle with the x-axis.

69. What angle does the line y equals square root of 3 times x make with the positive x axis?

70. What angle does the line y equals square root of 3 divided by 3 times x make with the positive x-axis?

71. Consider an equilateral triangle with each side having length 2k.

(a) What is the measure of each angle?

(b) Label one angle A. Drop a perpendicular from A to the side opposite A. Two 30 superscript circle angles are formed at A, and two right triangles are formed. What is the length of the sides opposite the 30 superscript circle angles?

(c) What is the length of the perpendicular in part (b)?

Image summary: This figure is a geometric diagram. It depicts a triangle where each of the three sides is labeled with the same algebraic expression. Because all three sides are equal in length, it can be concluded that the figure represents an equilateral triangle.

(d) From the results of parts (a)–(c), complete the following statement:

In a 30°–60° right triangle, the hypotenuse is always _ _ times as long as the shorter leg, and the longer leg has a length that is _ _ times as long as that of the shorter leg. Also, the shorter leg is opposite the _ _ angle, and the longer leg is opposite the _ _ angle.

72. Consider a square with each side of length k.

(a) Draw a diagonal of the square. What is the measure of each angle formed by a side of the square and this diagonal?

(b) What is the length of the diagonal?

(c) From the results of parts (a) and (b), complete the following statement:

In a 45 degree dash 45 degree right triangle, the hypotenuse has a length that is _ _ times as long as either leg.

Find the exact value of the variables in each figure.

Find a formula for the area of each figure in terms of s.

79. With a graphing calculator, find the coordinates of the point of intersection of the graphs of y equals x and y equals square root of 1 minus x squared. These coordinates are the cosine and sine of what angle between 0 degrees and 90 degrees?

80. Concept Check Suppose we know the length of one side and one acute angle of a 30 superscript circle-60 superscript circle right triangle. Is it possible to determine the measures of all the sides and angles of the triangle?

Figure 73 summary: This figure is a geometric diagram consisting of two adjacent right-angled triangles sharing a common vertical side. The diagram displays a large triangle divided by an altitude into two smaller right triangles, with labels for side lengths and interior angles. The left triangle features an acute angle and a hypotenuse, while the right triangle features a different acute angle and its own hypotenuse. The base is split into two segments, and the shared height is labeled. Based on the trigonometric relationships shown, the height of the shared side is shorter than the hypotenuse of the left triangle. The base segment of the left triangle is longer than the base segment of the right triangle, reflecting the difference in their respective interior angles.

Figure 75 summary: This figure consists of two geometric diagrams. Each diagram depicts a quadrilateral divided into two triangles by a diagonal line, with various interior angles and side lengths labeled using both numerical values and variables. In the first diagram, a right-angled triangle is paired with another triangle, showing a mix of acute and right angles. In the second diagram, a similar structure is shown where a right angle is formed between a side and a diagonal, and another right angle is present at a corner of the outer boundary. Based on the provided labels, it can be inferred that these figures are intended for geometric problem solving, requiring the application of trigonometric ratios and properties of triangles to determine the unknown side lengths represented by variables.

Figure 72 summary: This figure is a geometric diagram. It depicts a square with each of its four sides labeled with the variable k. The diagram illustrates a regular quadrilateral where all side lengths are equal, indicating that the perimeter is a multiple of the side length k.

Figure 74 summary: This figure is a geometric diagram of a triangle. The diagram depicts a large triangle divided into two smaller right-angled triangles by a vertical altitude. The labels indicate specific angle measurements and side lengths, including a known hypotenuse for the left triangle and interior angles for both sections. Based on the geometric properties shown, the altitude serves as a shared side between the two right triangles, allowing for the calculation of the remaining unknown side lengths and angles using trigonometric relationships.

Figure 77 summary: This figure consists of two separate diagrams. The first is a line chart depicting a sinusoidal wave passing through a rectangular frame, and the second is a geometric diagram of a right-angled triangle. The triangle is characterized by two equal side lengths and two equal acute angles. Based on the geometric properties shown, it can be inferred that the triangle is an isosceles right triangle, where the symmetry of the angles and side lengths confirms its specific classification.

Figure 78 summary: This figure is a geometric diagram. It depicts a triangle where all three interior angles are equal and all three side lengths are labeled with the same variable. Based on the identical angle measurements and side labels, it can be inferred that the figure represents an equilateral triangle.

Relating Concepts

For individual or collaborative investigation (Exercises 81 to 84)

The figure shows a 45 superscript circle central angle in a circle with radius 4 units. To find the coordinates of point P on the circle, work Exercises 81 to 84 in order.

81. Sketch a line segment from P perpendicular to the x-axis.

82. Use the trigonometric ratios for a 45 superscript circle angle to label the sides of the right triangle sketched in Exercise 81.

83. Which sides of the right triangle give the coordinates of point P? What are the coordinates of P?

84. The figure at the right shows a 60 superscript circle central angle in a circle of radius 2 units. Follow the same procedure as in Exercises 81 to 83 to find the coordinates of P in the figure.

2.2 Trigonometric Functions of Non-Acute Angles

- Reference Angles

- Special Angles as Reference Angles

- Determination of Angle Measures with Special Reference Angles

Image summary: This figure is a geometric diagram. It depicts a circle centered at the origin of a Cartesian coordinate system with a radius line extending from the center to a point labeled P on the circumference. The diagram includes a label for the radius length and an angle measurement between the radius and the positive x-axis. The point P is located in the first quadrant. The angle formed with the horizontal axis is an acute angle, and the distance from the center to the point P represents the radius of the circle. Based on the provided information, point P is positioned equidistant from the x and y axes because the angle is exactly half of a right angle. The position of point P can be determined using trigonometric functions based on the given radius and angle.

Reference Angles Associated with every nonquadrantal angle in standard position is an acute angle called its reference angle. A reference angle for an angle theta, written theta prime, is the acute angle made by the terminal side of angle theta and the x-axis.

Note Reference angles are always positive and are between 0 degrees and 90 degrees.

Figure 10 shows several angles theta (each less than one complete counterclockwise revolution) in quadrants 2, 3, and 4, respectively, with the reference angle theta prime also shown. In quadrant I, theta and theta prime are the same. If an angle theta is negative or has measure greater than 360 degrees, its reference angle is found by first finding its coterminal angle that is between 0 degrees and 360 degrees, and then using the diagrams in Figure 10.

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Figure 84 summary: This figure is a geometric diagram showing a circle on a Cartesian coordinate system. It depicts a circle centered at the origin with a point P located on the circumference in the first quadrant. A radius connects the origin to point P, with the angle between the radius and the positive x-axis labeled, and the length of the radius indicated. The diagram illustrates the relationship between polar coordinates and rectangular coordinates, demonstrating how a specific distance from the origin and a given angle determine the position of a point on a circle.