Trigonometric Functions Lesson

To define the six trigonometric functions, we will start with an angle in standard position, which we will name using the Greek letter theta θ. We can then choose some point P, with coordinates (x,y) on the terminal side of our angle θ. In this case, the actual coordinates of the point are (-6,6), but we will label the point generically as (x,y). We can then form a right triangle by drawing a vertical line segment from our point P to the x-axis. Now, we will define the distance from this point P with the coordinates of (x,y) to the origin with coordinates of (0,0) as r. To find the value of r or the distance from P(x,y) to (0,0), we can use the distance formula: $$d=\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ Let's replace d with r, x₁ with 0, and y₁ with 0: $$r=\sqrt{(x_2 - 0)^2 + (y_2 - 0)^2}$$ We can also replace x₂ with x and y₂ with y: $$r=\sqrt{x^2 + y^2}$$

The six Trigonometric Functions

The six trigonometric functions of angle θ are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). To define the six trigonometric functions, we let (x,y) represent some point other than the origin on the terminal side of an angle θ in standard position. The distance from our point (x,y) to the origin is defined as: $$r=\sqrt{x^2 + y^2}$$ Our six trigonometric functions are defined as: $$\text{sin}\hspace{.25em}θ=\frac{y}{r}$$ $$\text{cos}\hspace{.25em}θ=\frac{x}{r}$$ $$\text{tan}\hspace{.25em}θ=\frac{y}{x}, x ≠ 0$$ $$\text{csc}\hspace{.25em}θ=\frac{r}{y}, y ≠ 0$$ $$\text{sec}\hspace{.25em}θ=\frac{r}{x}, x ≠ 0$$ $$\text{cot}\hspace{.25em}θ=\frac{x}{y}, y ≠ 0$$ Let's look at an example.
Example #1: Use the given point on the terminal side of angle θ to find the value of cos θ. $$(-4,-3)$$ Let's use our formula for r: $$r=\sqrt{(-4)^2 + (-3)^2}$$ $$r=\sqrt{25}=5$$ $$x=-4$$ $$y=-3$$ Now we can plug into our formula for cos θ: $$\text{cos}\hspace{.25em}θ=\frac{x}{r}$$ $$\text{cos}\hspace{.25em}θ=\frac{-4}{5}=-\frac{4}{5}$$ Example #2: Use the given point on the terminal side of angle θ to find the value of sin θ. $$(\sqrt{13}, -6)$$ Let's use our formula for r: $$r=\sqrt{(\sqrt{13})^2 + (-6)^2}$$ $$r=\sqrt{49}=7$$ $$x=\sqrt{13}$$ $$y=-6$$ $$\text{sin}\hspace{.25em}θ=\frac{y}{r}$$ $$\text{sin}\hspace{.25em}θ=-\frac{6}{7}$$ Additionally, we may be asked to find the trigonometric function values of an angle when we know the equation of the line that coincides with the terminal ray. The equation of a line in standard form that passes through the origin: $$ax + by=0$$ If we were to restrict the x-values to either be: x ≤ 0 or x ≥ 0, we would get a ray with an endpoint at the origin. Let's look at an example.
Example #3: Find the values of the six trigonometric functions given the terminal side of θ.
Let's suppose we have an angle θ in standard position with the terminal side defined by: $$x - y=0, x ≥ 0$$ We can pick any point on the line except for (0,0) on the terminal side. If we choose an x-value of 3 and a y-value of 3, we can plug into our given formulas: $$r=\sqrt{3^2 + 3^2}$$ $$r=\sqrt{18}=3\sqrt{2}$$ $$x=3$$ $$y=3$$ $$\text{sin}\hspace{.25em}θ=\frac{3}{3\sqrt{2}}=\frac{\sqrt{2}}{2}$$ $$\text{cos}\hspace{.25em}θ=\frac{3}{3\sqrt{2}}=\frac{\sqrt{2}}{2}$$ $$\text{tan}\hspace{.25em}θ=\frac{3}{3}=1$$ $$\text{csc}\hspace{.25em}θ=\frac{3\sqrt{2}}{3}=\sqrt{2}$$ $$\text{sec}\hspace{.25em}θ=\frac{3\sqrt{2}}{3}=\sqrt{2}$$ $$\text{cot}\hspace{.25em}θ=\frac{3}{3}=1$$

Finding Function Values of Quadrantal Angles

When we have quadrantal angles, meaning angles with measures of multiples of 90°, we may encounter trigonometric functions that are undefined. This is due to the fact that in some cases, the function will show division by zero which is not defined. Let's look at an example.
Example #4: Find the values of the six trigonometric functions. Let's suppose we had a 90° angle. We can pick a point on the terminal side such as (0,1). Note that all x-coordinates would be 0. Let's find the value of our six trigonometric functions. $$r=\sqrt{1^2}=1$$ $$x=0$$ $$y=1$$ $$\text{sin}\hspace{.25em}θ=\frac{1}{1}=1$$ $$\text{cos}\hspace{.25em}θ=\frac{0}{1}=0$$ $$\text{tan}\hspace{.25em}θ=\frac{1}{0}\hspace{.2em}\text{undefined}$$ $$\text{csc}\hspace{.25em}θ=\frac{1}{1}=1$$ $$\text{sec}\hspace{.25em}θ=\frac{1}{0}\hspace{.2em}\text{undefined}$$ $$\text{cot}\hspace{.25em}θ=\frac{0}{1}=0$$

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