Lesson Explainer: Trigonometric Ratios on the Unit Circle Mathematics • 10th Grade

In this explainer, we will learn how to relate the π‘₯- and 𝑦-coordinates of points on the unit circle to trigonometric functions.

The unit circle is a circle with a radius of 1 whose center lies at the origin of a coordinate plane. For any point (π‘₯,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 πœƒ with the positive π‘₯-axis.

Using right triangle trigonometry, we can define the trigonometric functions in terms of the unit circle: sinoppositehypotenusesosincosadjacenthypotenusesocostanoppositeadjacentsotanπœƒ==𝑦1,𝑦=πœƒ,πœƒ==π‘₯1,π‘₯=πœƒ,πœƒ==𝑦π‘₯,𝑦π‘₯=πœƒ.

We note that tanπœƒ is not defined when π‘₯=0. We also observe that, while we have derived these definitions for an angle πœƒ in quadrant 1, they hold for an angle in any quadrant.

Theorem: Trigonometric Functions and the Unit Circle

The π‘₯- and 𝑦-coordinates of a point on the unit circle given by an angle πœƒ are defined by π‘₯=πœƒπ‘¦=πœƒ.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 sinπœƒ, given that πœƒ is in standard position and its terminal side passes through the point ο€Ό35,βˆ’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 π‘₯-axis. The angle is measured in a counterclockwise direction from the initial side to the terminal side. Hence, the angle πœƒ 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: ο€Ό35+ο€Ό45=𝑐,925+1625=𝑐,2525=𝑐.

Since 𝑐=1, 𝑐=1. This tells us that the point ο€Ό35,βˆ’45 lies on the unit circle. We recall that the π‘₯- and 𝑦-coordinates of a point on the unit circle given by an angle πœƒ are defined by π‘₯=πœƒπ‘¦=πœƒ.cosandsin

sinπœƒ is therefore equal to the value of the 𝑦-coordinate of the point, which is βˆ’45: sinπœƒ=βˆ’45

In addition to the standard trigonometric functions, it is also possible to define the reciprocal trigonometric functions (the reciprocal of a number π‘₯ is 1π‘₯). We can define them as follows.

Definition: Reciprocal Trigonometric Functions

For an angle πœƒβˆˆβ„, the reciprocal trigonometric functions are as follows:

  • The cosecant function: cscsinπœƒ=1πœƒ, for sinπœƒβ‰ 0
  • The secant function: seccosπœƒ=1πœƒ, for cosπœƒβ‰ 0
  • The cotangent function: cottanπœƒ=1πœƒ, for tanπœƒβ‰ 0

Since we can write the standard trigonometric function in terms of the unit circle, it is also possible for us to write the reciprocal functions in terms of the unit circle. That is, let us once more consider a point (π‘₯,𝑦) on the unit circle, with angle πœƒ to the positive π‘₯-axis.

Then, the reciprocal trigonometric functions can be written as follows: cscsinforseccosforcottanforπœƒ=1πœƒ=1𝑦,𝑦≠0,πœƒ=1πœƒ=1π‘₯,π‘₯β‰ 0,πœƒ=1πœƒ=π‘₯𝑦,𝑦≠0.

Let us consider an example where we use the unit circle to find the exact value of the secant function.

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 secπœƒ, given that πœƒ is in standard position and its terminal side passes through the point ο€Ό45,35.

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 π‘₯-axis. The angle is measured in a counterclockwise direction from the initial side to the terminal side. Hence, the angle πœƒ is as shown.

To calculate the value of secπœƒ, we will begin by determining whether the point with coordinates ο€Ό45,35 lies on the unit circle. To do this, we can consider a right triangle with side lengths of 45 units and 35 units as shown below.

Then, we can calculate the length of the hypotenuse, 𝑐, by using the Pythagorean theorem: ο€Ό45+ο€Ό35=𝑐,1625+95=𝑐,2525=𝑐.

Therefore, 𝑐=1, which means 𝑐=1. So, we have shown that the point ο€Ό45,35 lies on the unit circle.

The π‘₯- and 𝑦-coordinates of a point on the unit circle given by an angle πœƒ are defined by π‘₯=πœƒπ‘¦=πœƒ.cosandsin

We recall that seccosπœƒ=1πœƒ. In the case of the unit circle, secπœƒ=1π‘₯. secπœƒ=1=54.οŠͺ

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 π‘₯- or 𝑦-axis.

Example 3: Finding Cosine Values of Quadrantal Angles

Find the value of cos0.

Answer

The π‘₯- and 𝑦-coordinates of a point on the unit circle given by an angle πœƒ are defined by π‘₯=πœƒπ‘¦=πœƒ.cosandsin

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

The terminal side of the angle πœƒ=0radians lies on the π‘₯-axis, so the point at which it intersects the unit circle is (1,0). The π‘₯-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œ‹ That Have the Same Trigonometric Function

Suppose 𝑃 is a point on a unit circle corresponding to the angle of 4πœ‹3. Is there another point on the unit circle representing an angle in the interval [0,2πœ‹[ 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 4πœ‹3 radians with the positive π‘₯-axis measured in a counterclockwise direction. Since 4πœ‹3 is between πœ‹ and 3πœ‹2, we know that this point must lie in the third quadrant. Since both the π‘₯- and 𝑦-coordinates of this point are negative, we will define it as 𝑃(βˆ’π‘Ž,βˆ’π‘) for some positive constants π‘Ž and 𝑏.

We now recall the definition of the tangent function with respect to the unit circle. Given an angle πœƒ in standard position, where the coordinates of the point of intersection of the terminal side with the unit circle are (π‘₯,𝑦) and π‘₯β‰ 0, 𝑦π‘₯=πœƒ.tan

For point 𝑃, tan4πœ‹3=βˆ’π‘βˆ’π‘Ž=π‘π‘Ž.

The quotient of these π‘₯- 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 (π‘Ž,𝑏) which lies in the first quadrant.

We can map point 𝑃 onto point 𝑄 by performing a single rotation of πœ‹ radians. To find the value of 𝛼 such that tan𝛼=π‘π‘Ž, 4πœ‹3βˆ’πœ‹=πœ‹3.

Yes, there is another point on the unit circle that produces the same tangent value as the angle 4πœ‹3. The angle is πœ‹3.

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 βˆ π΄π‘‚π΅ in standard position intersects the unit circle 𝑂 at the point 𝐡 with coordinates ο€Ώ3√10,𝑦, where 𝑦>0. Find the value of sin𝐴𝑂𝐡.

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 π‘₯-axis. Since βˆ π΄π‘‚π΅ is in standard position and 𝐡 is not on the π‘₯-axis, the point 𝐴 must lie on the positive π‘₯-axis. We can therefore sketch βˆ π΄π‘‚π΅=πœƒ on the unit circle. Since both π‘₯- and 𝑦-coordinates are positive, point 𝐡 lies in the first quadrant.

We know that the π‘₯- and 𝑦-coordinates of a point on the unit circle given by an angle πœƒ are defined by π‘₯=πœƒπ‘¦=πœƒ.cosandsin

The value of sin𝐴𝑂𝐡 is therefore equal to the value of the 𝑦-coordinate of point 𝐡. By representing △𝐴𝑂𝐡 as a right triangle, we can find the value of 𝑦 by using the Pythagorean theorem.

We have 1=𝑦+ο€Ώ3√101=𝑦+910𝑦=110.

Since 𝑦>0 in this example, we obtain 𝑦=1√10 units.

The value of sin𝐴𝑂𝐡 is 1√10.

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 π‘₯- and 𝑦-coordinates of a point on the unit circle given by an angle πœƒ are defined by π‘₯=πœƒπ‘¦=πœƒ.cosandsin
  • The tangent ratio can also be defined for points on the unit circle (π‘₯,𝑦), where π‘₯β‰ 0: tanπœƒ=𝑦π‘₯.
  • The CAST diagram can be used to identify the signs of trigonometric function for angles in each quadrant.

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