Trigonometric Functions and the Unit Circle (Chapter 2.5)

These flashcards cover key concepts related to trigonometric functions and the unit circle as discussed in the lecture.

What is the unit circle defined as in this section?

The unit circle is a circle of radius one centered at the origin.

What is the sine of theta in terms of the unit circle?

The sine of theta is the y-coordinate of the point on the unit circle.

Define cosine of theta in relation to the unit circle.

The cosine of theta is the x-coordinate of the point on the unit circle.

What are the six trigonometric functions according to the unit circle?

Sine, cosine, tangent, cosecant, secant, cotangent.

What does the term 'coterminal angles' refer to?

Angles that differ by integer multiples of 2\pi and have the same sine and cosine values.

How does the definition of sine change when using the unit circle?

Sine equals b, where b is the y-coordinate of the point on the unit circle.

What is the period of the sine and cosine functions?

The period of sine and cosine is 2\pi .

What is the period of the tangent and cotangent functions?

The period of tangent and cotangent is \pi .

How are the coordinates for pi/6 on the unit circle represented?

The coordinates for pi/6 are (\frac{\sqrt{3}}{2}, \frac{1}{2}).

What relationship do the points on the unit circle have for angles with 6 in the denominator?

Angles with 6 in the denominator have coordinates that can be derived from pi/6 using sign adjustments based on the quadrant.

What are the even properties of the trig functions?

Cosine and secant are even functions, meaning \cos(-\theta) = \cos(\theta) and \sec(-\theta) = \sec(\theta).

What are the odd properties of the trig functions?

Sine, tangent, cosecant, and cotangent are odd functions, meaning f(-\theta) = -f(\theta).

What do we memorize for quickly computing the six trig functions?

We memorize specific angle values and their sine, cosine, and tangent relations, particularly for special angles.

What do the reference angles allow us to do?

Reference angles allow us to compute the values of trig functions for angles that are not in the first quadrant.

How do we handle trig functions of larger angles?

We reduce them using periodic properties by adding or subtracting multiples of their period.

What is the relationship between the sine and cosine functions with respect to the unit circle?

Sine and cosine values repeat every 2\pi , and are derived from the y and x coordinates of points on the unit circle respectively.

What is the significance of memorizing the unit circle?

Memorizing the unit circle allows for quick computation of trig function values without reference to triangles or tables.

What are the reciprocal identities for trigonometric functions?

\csc \theta = \frac{1}{\sin \theta}, \sec \theta = \frac{1}{\cos \theta}, \cot \theta = \frac{1}{\tan \theta} .

What are the quotient identities for tangent and cotangent?

\tan \theta = \frac{\sin \theta}{\cos \theta}, \cot \theta = \frac{\cos \theta}{\sin \theta} .

State the three Pythagorean Identities.

\sin^2 \theta + \cos^2 \theta = 1 , 1 + \tan^2 \theta = \sec^2 \theta , 1 + \cot^2 \theta = \csc^2 \theta .

Is the statement \sin(\theta + \phi) = \sin \theta + \sin \phi a trigonometric identity?

No, this is a common misconception (a non-identity).

Evaluate \sin(\frac{7\pi}{6}) .

-\frac{1}{2} .

Evaluate \cos(\frac{5\pi}{4}) .

-\frac{\sqrt{2}}{2} .

Evaluate \tan(\frac{2\pi}{3}) .

-\sqrt{3} .

Evaluate \sec(-\frac{\pi}{3}) .

2 .

Evaluate \csc(\frac{11\pi}{6}) .

-2 .

Evaluate \cot(\frac{5\pi}{4}) .

1 .

Evaluate \sin(\frac{13\pi}{4}) .

-\frac{\sqrt{2}}{2} .