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7. Properties of the Trigonometric Ratios
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Chapter 2
7. 

Properties of the Trigonometric Ratios

The sine and cosine of complementary angles share a unique bond. When exploring right triangles, it's evident that the sine of one acute angle is equivalent to the cosine of its complementary angle, and vice versa. This relationship is not just limited to sine and cosine; it extends to other trigonometric ratios like tangent and cotangent. Such relationships are pivotal in solving various mathematical problems, especially when dealing with angles and their complementary counterparts. Understanding these connections can be a powerful tool in the study of geometry and trigonometry, offering insights into the properties and behaviors of angles within triangles.
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12 Theory slides
8 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
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Properties of the Trigonometric Ratios
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In this lesson, the relationship between the sine and cosine of complementary angles will be investigated.

Catch-Up and Review

Here are a few recommended readings before getting started.

Explore

Investigating the Relationship Between Sine and Cosine

In the diagram's right triangles, some angle measures and side lengths are shown. Find the missing angle measures and lengths.

Three triangles with one common vertex
Use the information from diagram to complete the table of trigonometric ratios. Place the ratio into the appropriate cell.
Applet to fill in a table with sine and cosine ratios
Is there a recognizable pattern? If so, please describe it.
Discussion

Sine and Cosine of Complementary Angles

Consider the following table of trigonometric ratios for some acute angles.

As the measure of the angle increases, the sine ratio increases, and the cosine ratio decreases. Furthermore, some values are repeated. For example, the sine of is the same as the cosine of and the cosine of is the same as the sine of
This relationship leads to a rule.

The sine of an acute angle is equal to the cosine of its complementary angle. Similarly, the cosine of an acute angle is equal to the sine of its complementary angle.

Proof

By the Interior Angles Theorem, the sum of the interior angle measures of a triangle is For a right triangle, since one angle measures the other two angles are acute.

It follows that the sum of the measures of and is Therefore, they are complementary angles.
The sine and cosine ratios of complementary angles have a special relationship. To explore this, the three sides in the triangle will be labeled and
Using the definitions of sine and cosine, the following equations can be obtained.
By the Transitive Property of Equality, it can be said that and that

This is true for all pairs of complementary angles.

Example

Convert Between Sine and Cosine

Write the given expression in terms of cosine. Write your answer without the degree symbol.

Write the given expression in terms of sine. Write your answer without the degree symbol.

Hint

The sine of an acute angle is equal to the cosine of its complement. Similarly, the cosine of an acute angle is equal to the sine of its complement.

Solution
To write in terms of cosine, the relationship between the sine and cosine of complementary angles will be used. Recall that complementary angles add up to Therefore, to find the complement of subtract the given angle from
Substitute

SubTerm

Subtract term

Similarly, can be written in terms of sine.
Substitute

SubTerm

Subtract term

Pop Quiz

Practice Converting Between Sine and Cosine

Determine the value of that makes the equation true.

Example

Investigating Properties of Right Triangles

Since the sine and cosine ratios relate side lengths of right triangles, these ratios can help identify some properties of right triangles. Magdalena is curious to determine if a right triangle exists where the sine and cosine of one of its acute angles have the same value. To do so, she lets be the measure of the angle and writes the following equation.
She is considering drawing a diagram and using the definitions of sine and cosine.
a Use Magdalena's method to determine the type of the right triangle and the value of
b How can the sine and cosine of complementary angles be used to determine the value of

Answer
a Type of Right Triangle: Isosceles Right Triangle
Value of
b See solution.

Hint
a The sine of an acute angle is the ratio between the lengths of the opposite side and the hypotenuse. The cosine of an acute angle is the ratio between the lengths of the adjacent side and the hypotenuse.
b The complement of an angle that measures is

Solution
a Draw a right triangle with an acute angle that measures
The sine and cosine ratios of can be written as follows.
Since the above values can be substituted into this equation.
SubstituteII

,

MultEqn

It has been found that Therefore, is an isosceles triangle. By the Isosceles Triangle Theorem, and are equal to each other.
Since the sum of the interior angles of is the value of can be found.
SubstituteValues

Substitute values

▼
Solve for
AddTerms

Add terms

SubEqn

DivEqn

This means that the value of is
b Recall that the cosine of any acute angle is equal to the sine of its complementary angle. Since the complement of an angle that measures is the equation below holds true.
It is given that By substituting for the value of can be found.
For acute angles, the equation above is true when is equal to
AddEqn

DivEqn

Therefore, the value of is
Example

Solving Problems Using the Relationship Between Sine and Cosine

Let and be two acute angles of a right triangle. The sine of and the cosine of are expressed as follows.
What is the value of

Hint

Complementary angles add up to

Solution
Notice that and are complementary angles because they are acute angles of a right triangle.
Using the relationship between sine and cosine of complementary angles, the following equation can be written.
Finally, in the above equation, the given expressions for and can be substituted. By doing so, the value of can be found.
SubstituteExpressions

Substitute expressions

▼
Solve for
SubEqn

CommutativePropAdd

Commutative Property of Addition

FactorOut

Factor out

SubFrac

Subtract fractions

AddEqn

AddFrac

Add fractions

MoveRightFacToNumOne

MultEqn

The value of is 3.
Example

Using the Relationship Between Sine and Cosine to Solve Problems

In a right triangle, an acute angle measures and satisfies the following equation.
Find the value of

Hint

Use the sine and cosine relationship of complementary angles.

Solution
Recall that the cosine of an acute angle is equal to the sine of its complementary angle.
Since the complement of is is equal to
Because is an acute angle, then and are also acute angles. Since the sines of these two angles are equal, the angles have the same measure.
▼
Solve for
AddEqn

NumberToFrac

AddFrac

Add fractions

SubEqn

MultEqn

DivEqn

The value of is 56.
Discussion

Tangent and Cotangent of Complementary Angles

Like the sine and cosine, the same relationship exists between the tangent and cotangent.

The tangent of an acute angle is equal to the cotangent of its complementary angle. Similarly, the cotangent of an acute angle is equal to the tangent of its complementary angle. Therefore, the following statements hold true.

Proof

Consider a right triangle with side lengths and

right triangle
By using their definitions, the tangent and cotangent ratios can be written in terms of and
Since the acute angles of a right triangle are complementary, and are complementary angles. It can be seen that and This is true for all pairs of complementary angles.
Example

Solving Problems Using the Relationship Between Tangent and Cotangent

Find the value of the following expression.

Hint

Use the tangent and cotangent relationship of complementary angles.

Solution

Consider the product of the first term and the last terms of the expression.

Since the complement of is by the tangent and cotangent relationship of complementary angles, it can be said that is equal to
Substitute

Recall that the cotangent is the reciprocal of the tangent.

DenomMultFracToNumber

Therefore, the product of the first and the last terms is equal to Similarly, the product of the second and second to last terms is equal to and so on. The terms in this product can be grouped so that each pair has a product of Since the given product has terms, will stand alone.
Note that Furthermore, the rest of the factors can be grouped by pairs such that their product is also By the Identity Property of Multiplication, the value of the given expression is
Example

Calculating Distances Using Trigonometry

In physics, the phenomenon known as refraction of light is described as the change in a light's direction as it passes from one medium to another. Due to refraction, objects in the water may appear to be closer to the water's surface than they actually are. The diagram shows Ignacio's eye, from above the surface of the ocean, viewing a whale that looks to have an apparent depth of feet below the surface of the ocean.

External credits: @brgfx
If and calculate the distance between the apparent depth and actual depth of the whale. Round the answer to the nearest foot.

Hint

Use the tangent and cotangent relationship of complementary angles.

Solution

Begin by labeling points on the diagram.

External credits: @brgfx
Notice that and are collinear. Therefore, the measures of and add up to
SubstituteII

,

▼
Solve for
SubEqn

SubTerm

Subtract term

This means that and are complementary angles. Consequently, the cotangent of is equal to the tangent of its complementary angle.
From here, the distance between and can be calculated using tangent ratio of
Substitute the given value for and solve for
Substitute

▼
Solve for
MultEqn

DivEqn

Now that is found, can be calculated. By the Alternate Interior Angles Theorem, and are congruent.
External credits: @brgfx
The cotangent ratio of is then equal to Since is known, can be found.
Substitute

▼
Solve for
MultEqn

RearrangeEqn

Rearrange equation

RoundInt

Round to nearest integer

The whale is located at a depth of about feet. Therefore, Ignacio who is located at sees the whale about feet closer to the surface.
What a cool phenomenon made sense by complementary angles!
Closure

Secant and Cosecant Ratios

The relationship discussed throughout this lesson can be extended to include the secant and cosecant ratios.

As may have already been noticed, three of the trigonometric ratios start with the prefix
Consider an example trigonometric equation.
In this case, the prefix co denotes that is the co-angle, or complementary angle, of The identities seen in this lesson are referred to as cofunction identities.
Cofunction Identities



Level 1
Level 2
Level 3
Properties of the Trigonometric Ratios
Exercises

Solve the equation for

To solve this equation, we recall a property of the sine and cosine of complementary angles.

The sine of an acute angle equals the cosine of the complement of the angle.

Since complementary angles sum to 90^(∘), the sum of the sine's and cosine's arguments should equal 90^(∘). sin( x+30^(∘))=cos( 2x-60^(∘)) ⇓ ( x+30^(∘))+( 2x-60^(∘))=90^(∘) Let's solve for x by performing inverse operations until x is isolated.


As in Part A, we will equate the sum of the arguments to 90^(∘), then solve for x. sin( x/4+15^(∘))=cos( x+10^(∘)) ⇓ ( x/4+15^(∘))+( x+10^(∘))=90^(∘) We are ready to solve this equation for x.


Write the given expression in terms of cosine.

To write sin 67^(∘) in terms of cosine, we remember a property of the sine and cosine of complementary angles.

The sine of an acute angle is equal to the cosine of the angle's complementary angle.

Note that complementary angles sum to 90^(∘). This means we can write this property as the following equation. sin v = cos (90^(∘)-v) By substituting the measure of v, we can determine the corresponding expression for cosine.


As in Part A, we will substitute the given angle into the formula to determine the expression in terms of cosine.


Like in previous parts, we substitute ∠ v into the formula established in Part A then simplify to determine the corresponding expression in cosine.


If and what is
If and what is

To figure out the measure of v, let's recall the property of the sine and cosine of complementary angles.

The sine of an acute angle is the same as the cosine of the angle's complementary angle.

Notice that sin 30^(∘) and cos v have the same value. Therefore, 30^(∘) and v are, in fact, complementary angles. In other words, their sum is equal to 90^(∘). With this information, we can write and solve the following equation. 30^(∘)+ v=90^(∘) ⇔ v= 60^(∘) As we can see, v equals 60^(∘).

Like in Part A, we see that the cosine ratio equals the sine ratio. The sine ratio and the cosine ratio have the same value when the angles are complementary angles. That leads to the following equation which we can use to solve for v. 45^(∘)+ v=90^(∘) ⇔ v= 45^(∘)

Consider

Right triangle XYZ with sides y, z, and x
 Write an expression for
Write an expression for
Write an expression for
Write an expression for
Which of the trigonometric ratios from Part A to D are equal? Please explain your reasoning.

To obtain an expression for sin Z, we have to divide the angle's opposite side by the hypotenuse in the right triangle. sin Z = Opposite/Hypotenuse If we identify these sides relative to ∠ Z, we can write the expression.


This time, we want to determine an expression for cos Z. This means we must divide the angle's adjacent side by the hypotenuse. cos Z = Adjacent/Hypotenuse If we determine these sides relative ∠ Z, we can write a ratio for the cosine of Z.


As in Part A, we want to determine the sine value of the angle. However, this time we are looking for the sine value of ∠ Y. Therefore, XZ becomes the angle's opposite side.


As in Part B, we want to find the cosine value. Since we have changed the angle to ∠ Y, we get XY as adjacent side.


Let's summarize our findings from Part A to D. Part A: sin Z &= z/x [0.7em] Part B: cos Z &= y/x [0.7em] Part C: sin Y &= y/x [0.7em] Part D: cos Y &= z/x As we can see, sin Z and cos Y equal the same ratio as does cos Z and sin Y. Therefore, we can write the following equations. sin Z &= cos Y cos Z &= sin Y

Write the given expression in terms of sine.

In order to write cos 82^(∘) in terms of sine, we recall a property of the cosine and sine of complementary angles.

The cosine of an acute angle equals the sine of the angle's complementary angle.

Notice that the sum of complementary angles is 90^(∘). Therefore, we can express this property by the following equation. cos v = sin (90^(∘)-v) By substituting the measure of v, we can determine the corresponding expression for sine.


Just like in Part A, we will use the given angle in the formula to determine the equivalent expression in terms of sine.


As in Part A and B, we will substitute v into the equation relating sine and cosine, then simplify to determine the corresponding expression in terms of sine.


Properties of the Trigonometric Ratios

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