Sine and Cosine Function Values - AP PreCalculus Study Guide

The Core Idea: Sine and Cosine Function Values

This topic extends the definitions of sine and cosine beyond the unit circle to any angle in the $xy$-plane. The core idea is that the values of sine and cosine for an angle $\theta$ in standard position can be determined by the coordinates of any point $(x,y)$ that lies on the terminal ray of that angle. This is accomplished by forming a right triangle using the point $(x,y)$, the origin $(0,0)$, and the point $(x,0)$.

The distance from the origin to the point $(x,y)$, denoted by $r$, serves as the hypotenuse of this reference triangle. The sine and cosine functions are then defined as specific ratios of the $x$-coordinate, the $y$-coordinate, and this distance $r$. This framework allows us to calculate exact values for sine and cosine for angles in any quadrant, with the signs of the $x$ and $y$ coordinates directly determining the signs of the cosine and sine values, respectively.

Key Formulas

The definitions of sine and cosine for an angle $\theta$ in standard position, with a point $(x,y)$ on its terminal ray, are based on the following formulas derived directly from the geometry of the coordinate plane.

1. Distance from the Origin ($r$): The distance $r$ from the origin $(0,0)$ to the point $(x,y)$ is calculated using the distance formula, which is an application of the Pythagorean theorem. This value $r$ is always non-negative.

   $$r=\sqrt{x^2+y^2}$$

2. Sine of $\theta$: The sine of the angle $\theta$ is defined as the ratio of the $y$-coordinate to the distance $r$.

   $$\sin (\theta )=\frac{y}{r}$$

3. Cosine of $\theta$: The cosine of the angle $\theta$ is defined as the ratio of the $x$-coordinate to the distance $r$.

   $$\cos (\theta )=\frac{x}{r}$$

Understanding the Right Triangle and Quadrant Signs

The formulas for sine and cosine are fundamentally geometric. For any point $(x,y)$ on the terminal ray of an angle $\theta$, we can visualize a right triangle, often called a "reference triangle," that connects this point to the $x$-axis.

The vertices of this triangle are the origin $(0,0)$, the point $(x,y)$, and the point $(x,0)$.

• The side adjacent to the origin along the $x$-axis has a length of $|x|$.

• The side parallel to the $y$-axis has a length of $|y|$.

• The hypotenuse is the distance from the origin to $(x,y)$, which is $r$.

The crucial insight is that while the side lengths of the triangle are always positive, the coordinates $x$ and $y$ can be negative. Since $r$ is a distance, it is always positive ($r>0$ for any point not at the origin). Therefore, the signs of the sine and cosine ratios are determined entirely by the signs of the $y$ and $x$ coordinates, respectively. This dependence is dictated by the quadrant in which the terminal ray of $\theta$ lies.

• Quadrant I: $x>0$, $y>0$.

  • $\cos (\theta )=\frac{x}{r}=\frac{+}{+}⟹$ Cosine is positive.

  • $\sin (\theta )=\frac{y}{r}=\frac{+}{+}⟹$ Sine is positive.

• Quadrant II: $x<0$, $y>0$.

  • $\cos (\theta )=\frac{x}{r}=\frac{-}{+}⟹$ Cosine is negative.

  • $\sin (\theta )=\frac{y}{r}=\frac{+}{+}⟹$ Sine is positive.

• Quadrant III: $x<0$, $y<0$.

  • $\cos (\theta )=\frac{x}{r}=\frac{-}{+}⟹$ Cosine is negative.

  • $\sin (\theta )=\frac{y}{r}=\frac{-}{+}⟹$ Sine is negative.

• Quadrant IV: $x>0$, $y<0$.

  • $\cos (\theta )=\frac{x}{r}=\frac{+}{+}⟹$ Cosine is positive.

  • $\sin (\theta )=\frac{y}{r}=\frac{-}{+}⟹$ Sine is negative.

Core Concepts & Rules

• The sine and cosine values for any angle $\theta$ can be found using the coordinates of any point $(x,y)$ on the angle's terminal ray.

• The distance $r$ from the origin to the point $(x,y)$ is found using the formula $r=\sqrt{x^2+y^2}$.

• The value of $r$ represents the hypotenuse of the reference triangle and is always positive.

• The sine function is defined as the ratio $\sin (\theta )=\frac{y}{r}$. Its sign depends on the sign of the $y$-coordinate.

• The cosine function is defined as the ratio $\cos (\theta )=\frac{x}{r}$. Its sign depends on the sign of the $x$-coordinate.

• The quadrant of the terminal ray determines the signs of $x$ and $y$, and therefore the signs of $\sin (\theta )$ and $\cos (\theta )$.

Step-by-Step Example 1: Finding Sine and Cosine from a Point

Problem: The point $P(-5,12)$ lies on the terminal ray of an angle $\theta$ in standard position. Determine the exact values of $\sin (\theta )$ and $\cos (\theta )$.

Step 1: Identify the coordinates $x$ and $y$

From the given point $P(-5,12)$, we have:

• $x=-5$

• $y=12$

Step 2: Calculate the distance $r$

Use the formula $r=\sqrt{x^2+y^2}$.

$$r=\sqrt{(-5{)}^2+(12{)}^2}$$

$$r=\sqrt{25+144}$$

$$r=\sqrt{169}$$

$$r=13$$

Step 3: Calculate $\sin (\theta )$

Use the definition $\sin (\theta )=\frac{y}{r}$.

$$\sin (\theta )=\frac{12}{13}$$

Step 4: Calculate $\cos (\theta )$

Use the definition $\cos (\theta )=\frac{x}{r}$.

$$\cos (\theta )=\frac{-5}{13}$$

Final Answer:

$\sin (\theta )=\frac{12}{13}$ and $\cos (\theta )=-\frac{5}{13}$.

(Self-Check: The point $(-5,12)$ is in Quadrant II, where $x$ is negative and $y$ is positive. Our results show that $\cos (\theta )$ is negative and $\sin (\theta )$ is positive, which is consistent with the rules for Quadrant II.)

Step-by-Step Example 2: Exam-Style Application

Problem: If $\sin (\theta )=-\frac{8}{17}$ and the terminal ray of angle $\theta$ lies in Quadrant IV, what is the value of $\cos (\theta )$?

Step 1: Interpret the given information

We are given $\sin (\theta )=-\frac{8}{17}$. From the definition $\sin (\theta )=\frac{y}{r}$, we can associate $y=-8$ and $r=17$. We choose $r$ to be positive, as it represents a distance. The negative sign is associated with the $y$-coordinate.

• $y=-8$

• $r=17$

Step 2: Use the relationship $r=\sqrt{x^2+y^2}$ to find $x$

Substitute the known values of $y$ and $r$ into the equation.

$$17=\sqrt{x^2+(-8{)}^2}$$

Square both sides to eliminate the square root:

$$17^2=x^2+(-8{)}^2$$

$$289=x^2+64$$

Subtract 64 from both sides to solve for $x^2$:

$$x^2=289-64$$

$$x^2=225$$

Take the square root of both sides:

$$x=\pm \sqrt{225}$$

$$x=\pm 15$$

Step 3: Use the quadrant information to determine the sign of $x$

The problem states that the terminal ray of $\theta$ is in Quadrant IV. In Quadrant IV, the $x$-coordinate is positive. Therefore, we must choose the positive value for $x$.

$$x=15$$

Step 4: Calculate $\cos (\theta )$

Now that we have $x=15$ and $r=17$, we can use the definition $\cos (\theta )=\frac{x}{r}$.

$$\cos (\theta )=\frac{15}{17}$$

Final Answer:

$\cos (\theta )=\frac{15}{17}$.

Using Your Calculator

The concepts in this topic are fundamentally analytical, meaning they are solved using definitions and algebraic manipulation rather than direct calculator commands. You will not use a calculator to find the exact fractional answers required.

However, a calculator can be an excellent tool for checking your answer. The relationship $x^2+y^2=r^2$ leads directly to the fundamental Pythagorean identity ${\cos }^2(\theta )+{\sin }^2(\theta )=1$. You can use this to verify that your calculated sine and cosine values are a valid pair.

Checking Example 2:

In the second example, we were given $\sin (\theta )=-8/17$ and we found $\cos (\theta )=15/17$. To check if these values are correct, we can see if they satisfy the Pythagorean identity.

1. On your calculator's home screen, type the expression for ${\sin }^2(\theta )+{\cos }^2(\theta )$ using your values.

   $(-8/17{)}^2+(15/17{)}^2$

2. Press ENTER. The calculator should return $1$.

If the result is $1$, it confirms that your values for sine and cosine are consistent with each other. This does not check the quadrant signs, which you must do conceptually, but it verifies the magnitudes are correct.

AP Exam Quick Hit

Common Question Types

• Given a point on the terminal ray: You will be given a coordinate point, such as $(2,-5)$, and asked to find the exact value of $\sin (\theta )$ or $\cos (\theta )$.

  • Example: "The point $(2,-5)$ is on the terminal ray of an angle $\alpha$. What is the value of $\cos (\alpha )$?"

• Given one trigonometric ratio and a quadrant: You will be given the value of $\sin (\theta )$ or $\cos (\theta )$ and the quadrant of $\theta$, and you will be asked to find the value of the other function.

  • Example: "If $\cos (\beta )=-\frac{1}{3}$ and $\pi <\beta <\frac{3\pi }{2}$, what is the value of $\sin (\beta )$?"

• Conceptual sign analysis: You will be given information about the signs of $\sin (\theta )$ and $\cos (\theta )$ and asked to identify the quadrant.

  • Example: "If $\sin (\theta )<0$ and $\cos (\theta )>0$, in which quadrant does the terminal ray of angle $\theta$ lie?"

Common Mistakes

• Confusing $x$ and $y$ with $\cos (\theta )$ and $\sin (\theta )$: A frequent error is to state $\cos (\theta )=x$ and $\sin (\theta )=y$. This is only true on the unit circle where $r=1$. Always remember to divide by $r$.

• Incorrectly Assigning Signs: When solving for a missing coordinate (e.g., finding $x$ when given $y$ and $r$), students often forget to consider the given quadrant to choose the correct sign ($+$ or $-$) for the coordinate.

• Assuming $r$ can be negative: The value $r$ is a distance and must always be positive. When given a ratio like $\cos (\theta )=\frac{-3}{5}$, the negative sign must be associated with the $x$-coordinate ($x=-3$), not with $r$.

• Basic Algebraic Errors: Mistakes in squaring negative numbers (e.g., calculating $(-5{)}^2$ as $-25$ instead of $25$) will lead to an incorrect value for $r$ and wrong final answers.

• Mixing up Sine and Cosine Definitions: Accidentally using the $x$-coordinate for sine or the $y$-coordinate for cosine. Remember: Cosine goes with the $x$-coordinate, and sine goes with the $y$-coordinate.