Lesson Explainer: Trigonometric Ratios on the Unit Circle Mathematics

In this explainer, we will learn how to use the fact that the quadrant where an angle lies determines the signs of its sine, cosine, and tangent and solve trigonometric equations.

The unit circle is a circle with a radius of 1 whose center lies at the origin of a coordinate plane. For any point (𝑥,𝑦) on the unit circle, a right triangle can be formed as in the following diagram. The hypotenuse of this right triangle makes an angle 𝜃 with the positive 𝑥-axis.

Using right triangle trigonometry, we can define the trigonometric functions in terms of the unit circle: sinoppositehypotenusesosincosadjacenthypotenusesocostanoppositeadjacentsotan𝜃==𝑦1,𝑦=𝜃,𝜃==𝑥1,𝑥=𝜃,𝜃==𝑦𝑥,𝑦𝑥=𝜃.

We note that tan𝜃 is not defined when 𝑥=0. We also observe that, while we have derived these definitions for an angle 𝜃 in quadrant 1, they hold for an angle in any quadrant.

Theorem: Trigonometric Functions and the Unit Circle

The 𝑥- and 𝑦-coordinates of a point on the unit circle given by an angle 𝜃 are defined by 𝑥=𝜃𝑦=𝜃.cosandsin

In our first example, we will demonstrate how we can use these definitions of trigonometric functions in the unit circle to find exact values, given information about the terminal side of an angle.

Example 1: Finding the Value of a Trigonometric Function of an Angle given the Coordinates of the Point of Intersection of the Terminal Side and the Unit Circle

Find sin𝜃, given that 𝜃 is in standard position and its terminal side passes through the point 35,45.

Answer

An angle is said to be in standard position if the vertex is at the origin and the initial side lies on the positive 𝑥-axis. The angle is measured in a counterclockwise direction from the initial side to the terminal side. Hence, the angle 𝜃 is as shown.

We draw a right triangle with side lengths of 35 units and 45 units as demonstrated below.

Now, we can use the Pythagorean theorem to calculate the value of the missing dimension in the triangle: 35+45=𝑐,925+1625=𝑐,2525=𝑐.

Since 𝑐=1, 𝑐=1. This tells us that the point 35,45 lies on the unit circle. We recall that the 𝑥- and 𝑦-coordinates of a point on the unit circle given by an angle 𝜃 are defined by 𝑥=𝜃𝑦=𝜃.cosandsin

sin𝜃 is therefore equal to the value of the 𝑦-coordinate of the point, which is 45: sin𝜃=45

These definitions can also be applied when working with the reciprocal trigonometric functions. In example 2, we will use the equation of the unit circle to find an exact value of the secant function.

Consider a point (𝑥,𝑦) on the unit circle.

The right triangle formed will satisfy the Pythagorean theorem, such that 𝑥+𝑦=1.

The equation of the unit circle is therefore given by 𝑥+𝑦=1.

Example 2: Finding the Value of a Trigonometric Function of an Angle given the Coordinates of the Point of Intersection of the Terminal Side and the Unit Circle

Find sec𝜃, given that 𝜃 is in standard position and its terminal side passes through the point 45,35.

Answer

An angle is said to be in standard position if the vertex is at the origin and the initial side lies on the positive 𝑥-axis. The angle is measured in a counterclockwise direction from the initial side to the terminal side. Hence, the angle 𝜃 is as shown.

To calculate the value of sec𝜃, we will begin by determining whether the point with coordinates 45,35 lies on the unit circle. The equation of the unit circle is 𝑥+𝑦=1. Substituting 𝑥=45 and 𝑦=35 into the expression 𝑥+𝑦, 45+35=1625+925=1.

Since the values for 𝑥 and 𝑦 satisfy the equation 𝑥+𝑦=1, the point lies on the unit circle and we can use the following definition:

The 𝑥- and 𝑦-coordinates of a point on the unit circle given by an angle 𝜃 are defined by 𝑥=𝜃𝑦=𝜃.cosandsin

We recall that seccos𝜃=1𝜃. In the case of the unit circle, sec𝜃=1𝑥. sec𝜃=1=54.

We will now demonstrate how to find the exact value of a quadrantal angle, that is, an angle whose terminal side lies on either the 𝑥- or 𝑦-axis.

Example 3: Finding Cosine Values of Quadrantal Angles

Find the value of cos0.

Answer

The 𝑥- and 𝑦-coordinates of a point on the unit circle given by an angle 𝜃 are defined by 𝑥=𝜃𝑦=𝜃.cosandsin

The value of cos0 will therefore be the value of the 𝑥-coordinate at the point where the terminal side for 𝜃=0radians intersects the circumference of the unit circle, as shown on the diagram.

The terminal side of the angle 𝜃=0radians lies on the 𝑥-axis, so the point at which it intersects the unit circle is (1,0). The 𝑥-coordinate, and hence the value of cos0, is 1.

The value of cos0 is 1.

In our previous examples, we showed how to use the definition of the unit circle to find the exact value of trigonometric functions. In our next example, we will demonstrate how the unit circle validates the periodicity of such functions.

Example 4: Exploring the Different Angles between 0 and 2𝜋 That Have the Same Trigonometric Function

Suppose 𝑃 is a point on a unit circle corresponding to the angle of 4𝜋3. Is there another point on the unit circle representing an angle in the interval [0,2𝜋[ that has the same tangent value? If yes, give the angle.

Answer

We will begin by sketching the unit circle and the point 𝑃 which creates an angle of 4𝜋3 radians with the positive 𝑥-axis measured in a counterclockwise direction. Since 4𝜋3 is between 𝜋 and 3𝜋2, we know that this point must lie in the third quadrant. Since both the 𝑥- and 𝑦-coordinates of this point are negative, we will define it as 𝑃(𝑎,𝑏) for some positive constants 𝑎 and 𝑏.

We now recall the definition of the tangent function with respect to the unit circle. Given an angle 𝜃 in standard position, where the coordinates of the point of intersection of the terminal side with the unit circle are (𝑥,𝑦) and 𝑥0, 𝑦𝑥=𝜃.tan

For point 𝑃, tan4𝜋3=𝑏𝑎=𝑏𝑎.

The quotient of these 𝑥- and 𝑦-coordinates is positive. We now observe that there must be a second point on the unit circle for which this is the case. This is the point with coordinates (𝑎,𝑏) which lies in the first quadrant.

We can map point 𝑃 onto point 𝑄 by performing a single rotation of 𝜋 radians. To find the value of 𝛼 such that tan𝛼=𝑏𝑎, 4𝜋3𝜋=𝜋3.

Yes, there is another point on the unit circle that produces the same tangent value as the angle 4𝜋3. The angle is 𝜋3.

In our previous example, we used the unit circle to demonstrate the periodicity of the tangent function. We can use a similar process to help us determine the sign of any trigonometric functions given the size of their angle.

Consider point 𝑃 located in the first quadrant.

We observe the value of the 𝑥- and 𝑦-coordinates, and thus the values of cos𝜃 and sin𝜃, to be positive. Further, since tan𝜃=𝑦𝑥, tan𝜃 is also positive in this quadrant.

Now, consider the point 𝑃 located in the second quadrant.

The value of the 𝑥-coordinate, and hence the value of cos𝜃, is negative in this quadrant. Since the 𝑦-coordinate is positive here, the value of sin𝜃 is positive in this quadrant. In a similar way, we observe the tangent function for values of 𝜃 in quadrant 2 to be the quotient of a positive and negative number. tan𝜃 is therefore negative in this quadrant.

We can repeat this process in quadrants 3 and 4 to find the following:

  • In quadrant 3, tan𝜃 is positive and sin𝜃 and cos𝜃 are negative.
  • In quadrant 4, cos𝜃 is positive and sin𝜃 and tan𝜃 are negative.

This is simplified in the CAST diagram. The letters in the acronym CAST indicate the trigonometric functions that are positive in each quadrant.

Definition: The CAST Diagram

  • In quadrant 1, All values are positive,
  • In quadrant 2, Sine is positive,
  • In quadrant 3, Tangent is positive,
  • In quadrant 4, Cosine is positive.

Let us demonstrate how the CAST diagram can help us simplify a problem by considering the expression sin7𝜋6. Drawing the angle to be 7𝜋6 radians on the unit circle, we see that this angle lies in the third quadrant.

The CAST diagram tells us that the tangent function is positive in this quadrant, while the sine and cosine functions are negative. We can therefore infer that the value of sin7𝜋6 will be negative.

As demonstrated below, a rotation about the origin by 𝜋 radians has the same effect as switching the signs of the 𝑥- and 𝑦-coordinates. Using the relation between the 𝑥-coordinates of points on the unit circle and its standard angle, we obtain the identity sinsin(𝜃)=(𝜃𝜋).

Hence, returning to our example, sinsinsin7𝜋6=7𝜋6𝜋=𝜋6.

Since sin𝜋6=12, this means that sin7𝜋6=12. As predicted by the CAST diagram, we note that sin7𝜋6 takes a negative value.

In our final example, we will demonstrate how to use the unit circle to evaluate a simple trigonometric function.

Example 5: Finding the Value of a Trigonometric Function given the Coordinates of the Point of Intersection of the Unit Circle with the Terminal Side of an Angle in Standard Position

The terminal side of 𝐴𝑂𝐵 in standard position intersects the unit circle 𝑂 at the point 𝐵 with coordinates 310,𝑦, where 𝑦>0. Find the value of sin𝐴𝑂𝐵.

Answer

An angle is said to be in standard position if the vertex is at the origin and the initial side lies on the positive 𝑥-axis. Since 𝐴𝑂𝐵 is in standard position and 𝐵 is not on the 𝑥-axis, the point 𝐴 must lie on the positive 𝑥-axis. We can therefore sketch 𝐴𝑂𝐵=𝜃 on the unit circle. Since both 𝑥- and 𝑦-coordinates are positive, point 𝐵 lies in the first quadrant.

We know that the 𝑥- and 𝑦-coordinates of a point on the unit circle given by an angle 𝜃 are defined by 𝑥=𝜃𝑦=𝜃.cosandsin

The value of sin𝐴𝑂𝐵 is therefore equal to the value of the 𝑦-coordinate of point 𝐵. By representing 𝐴𝑂𝐵 as a right triangle, we can find the value of 𝑦 by using the Pythagorean theorem.

We have 1=𝑦+3101=𝑦+910𝑦=110.

Since 𝑦>0 in this example, we obtain 𝑦=110 units.

The value of sin𝐴𝑂𝐵 is 110.

Let us finish by recapping some key concepts from this explainer.

Key Points

  • The unit circle is a circle with a radius of 1 centered at the origin of coordinate plane.
  • The 𝑥- and 𝑦-coordinates of a point on the unit circle given by an angle 𝜃 are defined by 𝑥=𝜃𝑦=𝜃.cosandsin
  • The tangent ratio can also be defined for points on the unit circle (𝑥,𝑦), where 𝑥0: tan𝜃=𝑦𝑥.
  • The CAST diagram can be used to identify the signs of trigonometric function for angles in each quadrant.

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