**Introduction to Trigonometric Functions**

There are six trigonometric functions, but four of them are written in terms of two of the main functions—sine and cosine. Although trigonometry was developed to solve problems involving triangles, there is a very close relationship between sine and cosine and the unit circle. Suppose an angle *θ* is given. The *x* -coordinate of the point on the unit circle for *θ* is cosine of the angle (written cos *θ* ). The *y* -coordinate of the point is sine of the angle (written sin *θ* ). For example, suppose the point determined by the angle *θ* is (3/5, 4/5). Then cos *θ* = 3/5 and sin *θ* = 4/5. See Figure 13.9.

**Examples**

Find sin *θ* and cos *θ*.

sin and cos *θ* = −1/2

sin and cos

**Sine and Cosine**

The equation for the unit circle is *x* ^{2} + *y* ^{2} = 1. For an angle *θ* , we can replace *x* with cos *θ* and *y* with sin *θ* . This changes the equation to cos ^{2} *θ* + sin ^{2} *θ* = 1 (cos ^{2} *θ* means (cos *θ* ) ^{2} and sin ^{2} *θ* means (sin *θ* ) ^{2} ). This is an important equation. It allows us to find cos *θ* if we know sin *θ* and sin *θ* if we know cos *θ* . Solving this equation for cos *θ* gives us . Solving it for sin *θ* gives us .. For example, if we know sin , we can find cos *θ* .

Is cos We cannot answer this without knowing where *θ* is. If we know that *θ* is in Quadrants I or IV, then cos because cosine is positive in Quadrants I and IV. If we know that *θ* is in Quadrants II or III, then cos because cosine is negative in Quadrants II and III.

**Examples**

Find sin *θ* and cos *θ* .

- The terminal point for
*θ*is (−12/13,*y*), and*θ*is in Quadrant II.

Because the *y* -values in Quadrant II are positive, sin *θ* is positive.

- The terminal point for
*θ*is (*x*, −1/9), and*θ*is in Quadrant III. -
Both sine and cosine are negative in Quadrant III, so we will use the negative square root. Using sin

*θ*= −1/9, we have

**Sine and Cosine Values for Common Angles**

The values for sine and cosine of the following angles should be memorized: 0, *π* /6, *π* /4, *π* /3, and *π* /2. See Figure 13.12.

**Sine and Cosine Values in the Four Quadrants**

All of these angles are also reference angles in the other three quadrants. You should either memorize or be able to quickly compute them. The information is in the table below.

**Other Trigonometric Functions - Tangent, Cotangent, Secant, and Cosecant**

The other trigonometric functions are tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). All of them can be written as a ratio with sine, cosine, or both.

Sine and cosine can be evaluated at any angle. This is not true for the other trigonometric functions. For example tan *π* /2 = sin *π* /2/cos *π* /2 and sec *π* /2 = 1/cos *π* /2 are not defined because cos *π* /2 = 0. We can find all six trigonometric functions for an angle *θ* if we either know both coordinates of its terminal point or if we know one coordinate and the quadrant where *θ* lies.

Before we begin the next set of problems, we will review a shortcut that will save some computation for tan *θ* . A compound fraction of the form *(a/b)/(c/b)* simplifies to *a/c* .

**Examples**

Find all six trigonometric functions for *θ* .

- The terminal point for
*θ*is (24/25, 7/25)

*θ*=*π*/3

*θ*= 5*π*/6

- The
*x*-coordinate of*θ*is 2/5, and*θ*is in Quadrant IV.

Practice problems for this concept can be found at: Trigonometry Practice Test.

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