Introduction to Trigonometric Functions
This is it! The moment you’ve been waiting for! We begin our foray into trigonometry. If you ask, ‘what is the purpose of precalculus?’, we’d reply that Chapter 2 illustrates one of the primary uses of this branch of mathematics. Using trigonometry, we can solve for sides of triangles – or distances between real world artifacts – by performing calculations with angles. In fact, our introductory problem provides a great example of the usefulness of precalculus:
You take your cousin Mo downtown to visit the Institute of Texan Cultures and sightsee in HemisFair Park. Because Mo is from New Jersey, he’s dying to ride the elevator up the Tower of the Americas for an unobscured view of the horizon. You part with Mo, who heads towards the Tower while you grab an aquas frescas to cool off. As you take your last sip of this sweet nectar from the gods, Mo calls you from the top of the tower and wants you to join him. If the tower is 750 feet tall and, according to your inclinometer the angle of elevation from you to the top is 17o, how far of a walk do you have to meet your cousin? Do you go meet him? Explain why you do or don’t walk to the tower.
SOLUTION: The tangent function of an angle in a right triangle is equal to the ratio of side opposite to the side adjacent. In this problem the Tower of the Americas forms a right triangle with the horizon line and the angle of inclination which extends to the top of the tower. This means, the distance from you to the tower can be calculated as tangent (17o) is equal to 750 feet divided by [latex]x[/latex] feet (the distance). Putting this into a calculator yields a distance of 2,453.14 feet, which is approximately equivalent to half a mile. Are you up for a half mile walk? My answer depends on the time of year and the temperature outside (n.b. you did just drink an aquas frescas)!
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In Chapter 2 we will begin our study of trigonometry with a discussion of angles and an introduction to the Unit Circle (the most sacred circle in precalculus!). The unit circle is used to reference angles to evaluate trigonometric functions. In this module, you will explore the properties of key angles in all quadrants of the unit circle measured in both degrees and radians to compute the values of the six trigonometric functions: sine, cosine, tangent, secant, cosecant, and cotangent functions. You will also have the opportunity to solve for a coterminal angle, evaluate sine and cosine functions at given angles, apply the formula for the length of an arc subtended by an angle in a circle of radius r, and solve linear and angular velocity scenarios by applying your knowledge of the unit circle. Finally, you will explore right triangle trigonometry to define the six trigonometric functions in terms of the lengths of the sides of a right triangle and solve for the sides of a right triangle when given an angle and the length of one side.
A complete list of the Key Takeaways from this chapter are included in the table below:
Key Takeaways
- Define an angle in standard position and know when its measure is positive or negative.
- Know the relationship between degrees and radians and be able to sketch angles of any measure.
- Recall formulas for finding the length of a circular arc and the area of a sector of a circle.
- Relate the linear speed of an object in circular motion with its angular speed in either radians per unit of time or revolutions per unit of time (or vice versa).
- Determine the period of the six trigonometric functions and their signs in a given quadrant of the Cartesian plane.
- Recall the reciprocal and quotient identities based on the definitions from the Unit Circle of the six trigonometric functions.
- Solve for the value of an x or y coordinate given one of the values as a point on the unit circle and a quadrant.
- Use the Unit Circle to derive the three Pythagorean Identities to complete the set of Fundamental Identities.
- Find the exact value of the remaining trigonometric functions given the value of one and the sign of another using either a circle of radius r or the Fundamental Identities.
- Determine and use even-odd properties to find exact values for the six trigonometric functions.
- Apply the definitions of the six trigonometric functions to solve any right triangle given or sketched from application questions.
