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Similarity, Proof, and Trigonometry

Properties of the Trigonometric Ratios

In this lesson, the relationship between the sine and cosine of complementary angles will be investigated.

Catch-Up and Review

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.

Use the information from diagram to complete the table of trigonometric ratios. Place the ratio into the appropriate cell.

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
Similarly, can be written in terms of sine.

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.
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.
Solve for
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
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.
Solve for
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

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
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
Recall that the cotangent is the reciprocal of the tangent.

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.

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.

Notice that and are collinear. Therefore, the measures of and add up to
Solve for
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
Solve for
Now that is found, can be calculated. By the Alternate Interior Angles Theorem, and are congruent.
The cotangent ratio of is then equal to Since is known, can be found.
Solve for
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
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