25:35
Trig Definitions, Continued
LP W
Overview
This video continues the introduction to trigonometric functions by defining tangent, cotangent, secant, and cosecant. It builds upon the previous lesson's definitions for sine and cosine, emphasizing that all trigonometric functions take angles in standard position as input and output specific ratios. The video uses a consistent example with a point (3, 4) and a radius of 5 to illustrate how these ratios are calculated for angles terminating in different quadrants. It also highlights the reciprocal relationships between trigonometric function pairs (sine/cosecant, cosine/secant, tangent/cotangent) and how the signs of these functions vary across the four quadrants based on the signs of the x and y coordinates.
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Chapters
- Trigonometric functions (sine through secant) take angles measured from standard position as input.
- Standard position means angles are measured from the positive x-axis.
- Counterclockwise measurement yields positive angles; clockwise yields negative angles.
- The definition of sine is the ratio y/r, and cosine is x/r, where (x, y) is a point on the terminal side of the angle and r is the distance from the origin to that point.
Understanding the input (angles in standard position) and the basic definitions of sine and cosine is crucial for correctly applying these functions in various mathematical contexts.
Using a point (3, 4) with r=5, sine is calculated as 4/5 and cosine as 3/5.
- Tangent is defined as the ratio y/x (opposite over adjacent).
- Cotangent is defined as the ratio x/y (adjacent over opposite), making it the reciprocal of tangent.
- The signs of tangent and cotangent in each quadrant depend on the signs of x and y.
- Tangent is positive in Quadrants I (y>0, x>0) and III (y<0, x<0), and negative in Quadrants II (y>0, x<0) and IV (y<0, x>0).
These definitions extend the trigonometric toolkit, allowing us to analyze angles and their relationships in triangles and the unit circle using ratios involving the x and y coordinates.
For an angle terminating in Quadrant II with point (3, 4) and r=5, tangent is 4/(-3) = -4/3, and cotangent is (-3)/4 = -3/4.
- Cosecant is defined as the ratio r/y, making it the reciprocal of sine (y/r).
- Secant is defined as the ratio r/x, making it the reciprocal of cosine (x/r).
- The signs of cosecant and secant mirror those of sine and cosine, respectively, because 'r' is always positive.
- Cosecant has the same sign as sine (positive in Quadrants I and II, negative in III and IV).
- Secant has the same sign as cosine (positive in Quadrants I and IV, negative in II and III).
Understanding cosecant and secant as reciprocals of sine and cosine simplifies calculations and provides complementary perspectives on trigonometric relationships.
Using the same example with r=5, if sine is 4/5, cosecant is 5/4. If cosine is 3/5, secant is 5/3.
- The sign of each trigonometric function in a given quadrant is determined by the signs of the x and y coordinates (and r, which is always positive).
- Quadrant I: All functions are positive (x>0, y>0).
- Quadrant II: Sine and cosecant are positive (y>0); cosine, secant, tangent, and cotangent are negative (x<0).
- Quadrant III: Tangent and cotangent are positive (x<0, y<0); sine, cosecant, cosine, and secant are negative.
- Quadrant IV: Cosine and secant are positive (x>0); sine, cosecant, tangent, and cotangent are negative (y<0).
Knowing the sign patterns of trigonometric functions in each quadrant is essential for solving trigonometric equations and understanding the behavior of trigonometric graphs.
A mnemonic device like 'All Students Take Calculus' can help remember which functions are positive in each quadrant (All in I, Sine in II, Tangent in III, Cosine in IV).
- The core definition of trigonometric functions relies on choosing any point (other than the origin) on the terminal side of an angle in standard position.
- This definition consistently produces the same ratios regardless of the point chosen on the terminal side.
- The concept of the unit circle (radius = 1) is a simplification that arises from this definition, making calculations easier.
- Future lessons will explore why the unit circle is particularly useful.
Recognizing that the fundamental definition holds true for any point on the terminal side justifies the use of the unit circle as a powerful tool for understanding and calculating trigonometric values.
The video used a point (3, 4) with r=5, but the ratios would be the same if a point like (6, 8) with r=10 were used, or if scaled down to (0.6, 0.8) with r=1 on a unit circle.
Key takeaways
- All six trigonometric functions take angles in standard position as their input.
- Sine, cosine, tangent, cotangent, secant, and cosecant are defined as specific ratios of x, y, and r (where r is the distance from the origin to a point (x, y) on the terminal side of the angle).
- Tangent and cotangent are defined using x and y coordinates (y/x and x/y, respectively).
- Cosecant and secant are defined using r and y or x (r/y and r/x, respectively), making them reciprocals of sine and cosine.
- The sign of a trigonometric function in a specific quadrant depends on the signs of x and y in that quadrant.
- Understanding the reciprocal relationships simplifies remembering and calculating function values.
- The fundamental definition of trigonometric functions is robust and applies regardless of the specific point chosen on the terminal side of an angle.
Key terms
Standard PositionTerminal SideAngle MeasurementTangent (tan)Cotangent (cot)Secant (sec)Cosecant (csc)RatioReciprocal FunctionsQuadrant Signs
Test your understanding
- How are the inputs for all six trigonometric functions measured?
- What are the definitions of tangent and cotangent in terms of x, y, and r?
- Explain the reciprocal relationship between sine and cosecant, and cosine and secant.
- How does the sign of the tangent function change as an angle's terminal side moves from Quadrant II to Quadrant III?
- Why is the choice of a specific point on the terminal side of an angle not critical for defining trigonometric ratios?