7.2
Trigonometric Functions of Real Numbers
The unit circle—a circle with a radius of one, centered at the origin of the coordinate plane—serves as the foundational framework for defining trigon…
The unit circle, centered at the origin, defines the trigonometric functions.
Each real number represents an arc length measured from the positive x-axis and rotate it counterclockwise around the circle.
The x-coordinate gives the cosine of the real number. Similarly, the y-coordinate gives the sine.
Tangent is derived as sine over cosine, cotangent is cosine over sine, secant is the reciprocal of cosine, and cosecant is the reciprocal of sine.
Dividing the circle into four quadrants helps identify a function’s sign.
All functions are positive in the first quadrant, where x and y are positive. In the second, since x is negative, the cosine is negative. As y is positive, sine is also positive. Tangent, which is y over x, is negative because a positive over a negative is negative.
In the third, tangent and cotangent are positive, as x and y are negative. In the fourth, only cosine and secant are positive, as x is positive and y is negative.
Trigonometric functions are useful for real-world calculations, such as estimating a building’s height by multiplying the known horizontal distance by the tangent of the known angle of inclination.
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Q1: How does the unit circle define trigonometric functions?
The unit circle, centered at the origin with radius one, provides the geometric foundation for trigonometric functions. Each real number represents an arc length measured counterclockwise from the positive x-axis. The x-coordinate of a point on the circle equals cosine, while the y-coordinate equals sine. All other trigonometric functions derive from these two fundamental ratios.
Q2: What are the six trigonometric functions and how are they related?
The six trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. Tangent is sine divided by cosine, while cotangent is cosine divided by sine. Secant is the reciprocal of cosine, and cosecant is the reciprocal of sine. These derived functions extend the foundational sine and cosine relationships.
Q3: Why do trigonometric function signs change across different quadrants?
The unit circle's four quadrants determine function signs based on coordinate values. In quadrant one, all functions are positive. In quadrant two, only sine and cosecant are positive because x is negative. In quadrant three, tangent and cotangent are positive since both x and y are negative. In quadrant four, only cosine and secant are positive.
Q4: What are even and odd properties of trigonometric functions?
Trigonometric functions exhibit symmetry properties derived from the unit circle. Cosine and secant are even functions, satisfying cos(−t) = cos(t). Sine, tangent, cotangent, and cosecant are odd functions, satisfying sin(−t) = −sin(t). These properties reflect the geometric symmetry of the circle about the x-axis and origin.
Q5: How can trigonometric functions solve real-world measurement problems?
Trigonometric functions enable practical calculations like estimating building heights. By multiplying a known horizontal distance by the tangent of the angle of inclination, you can determine vertical height. This application demonstrates how trigonometric functions problem solving connects geometric relationships on the unit circle to tangible engineering and surveying tasks.
Q6: What is the relationship between arc length and real numbers on the unit circle?
On the unit circle, each real number corresponds to an arc length measured counterclockwise from the positive x-axis. This geometric representation allows real numbers to be visualized as distances along the circle's circumference. The arc length directly determines the coordinates of the point, which define the sine and cosine values for that real number.
Q7: How do you determine which trigonometric functions are positive in each quadrant?
Use the coordinate signs in each quadrant to determine function positivity. Since sine equals the y-coordinate and cosine equals the x-coordinate, their signs depend on quadrant location. Tangent, being y over x, is positive when both coordinates share the same sign. Reciprocal functions follow their corresponding primary functions' signs across all quadrants.