Trigonometric Functions and Their Properties

This lesson explores the graphs of four trigonometric functions: tangent, cotangent, cosecant, and secant, by relating them to the sine and cosine graphs. It covers the domains, reciprocal pairs, and graphing techniques, including identifying zeros and asymptotes. The lesson also discusses transformations using parameters a, b, h, and k, and provides tips for understanding the behavior of these graphs, such as the period and direction of curves.

What is the relationship between the tangent and cosine graphs?

• Tangent is sine divided by cosine.

• Tangent is the derivative of cosine.

• Tangent is the reciprocal of cosine.

• Tangent is cosine divided by sine.

Where is the tangent function undefined?

• Where both sine and cosine are zero.

• Where neither sine nor cosine is zero.

• Where cosine is zero.

• Where sine is zero.

In which quadrants is the tangent function positive?

• Quadrants 1 and 3

• Quadrants 2 and 3

• Quadrants 3 and 4

• Quadrants 1 and 2

What is the period of the tangent function?

• 2π

• π

• π/2

• 4π

How does the cotangent graph differ from the tangent graph?

• Cotangent decreases from left to right.

• Cotangent is the reciprocal of sine.

• Cotangent is always increasing.

• Cotangent has a period of 2π.

Where does the cotangent function have asymptotes?

• Where neither sine nor cosine is zero.

• Where cosine is zero.

• Where sine is zero.

• Where both sine and cosine are zero.

What is the period of the cosecant function?

• π/2

• 4π

• 2π

• π

Where does the secant function have asymptotes?

• Where both sine and cosine are zero.

• Where neither sine nor cosine is zero.

• Where cosine is zero.

• Where sine is zero.

How does a negative 'a' value affect the graph of a trigonometric function?

• It shifts the graph vertically.

• It reflects the graph over the x-axis.

• It shrinks the graph horizontally.

• It stretches the graph vertically.

What is the effect of horizontal stretching on the period of tangent and cotangent?

• It halves the period.

• It doubles the period.

• It changes the period to π over b.

• It changes the period to 2π over b.