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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Lesson Settings & Tools
| | 12 Theory slides |
| | 8 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Properties of the Trigonometric Ratios
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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.
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Discussion
Sine and Cosine of Complementary Angles
Consider the following table of trigonometric ratios for some acute angles.
| sin θ | cos θ | |
|---|---|---|
| θ = 15^(∘) | 0.258819 ... | 0.965925 ... |
| θ = 30^(∘) | 0.5 | 0.866025 ... |
| θ = 45^(∘) | 0.707106 ... | 0.707106 ... |
| θ = 60^(∘) | 0.866025 ... | 0.5 |
| θ = 75^(∘) | 0.965925 ... | 0.258819 ... |
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 30^(∘) is the same as the cosine of 60^(∘), and the cosine of 75^(∘) is the same as the sine of 15^(∘). cllll sin 30^(∘) & = & cos 60^(∘) & = & 0.5 [0.8em] cos 75^(∘) & = & sin 15^(∘) & = & 0.258819... 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.
sin θ =cos(90^(∘) -θ) cos θ =sin(90^(∘) -θ)
Proof
By the Interior Angles Theorem, the sum of the interior angle measures of a triangle is 180^(∘). For a right triangle, since one angle measures 90^(∘), the other two angles are acute.
It follows that the sum of the measures of ∠ A and ∠ B is 90^(∘). Therefore, they are complementary angles. m∠ A + m∠ B =90^(∘) The sine and cosine ratios of complementary angles have a special relationship. To explore this, the three sides in the triangle will be labeled x, y, and z.
Using the definitions of sine and cosine, the following equations can be obtained.
sin A=y/z & & cos A=x/z [1em]
sin B=x/z & & cos B=y/z
By the Transitive Property of Equality, it can be said that sin A=cos B and that cos A =sin B.
sin A= y/z cos B= y/z ⇒ sin A = cos B [3em] cos A= x/z sin B= x/z ⇒ cos A= sin B
This is true for all pairs of complementary angles.
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Example
Convert Between Sine and Cosine
Write the given expression in terms of cosine. Write your answer without the degree symbol.
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Write the given expression in terms of sine. Write your answer without the degree symbol.
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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 sin 27^(∘) in terms of cosine, the relationship between the sine and cosine of complementary angles will be used. Recall that complementary angles add up to 90^(∘). Therefore, to find the complement of 27^(∘), subtract the given angle from 90^(∘).
sin θ = cos(90^(∘) -θ)
Substitute
θ= 27^(∘)
sin 27^(∘) = cos(90^(∘)- 27^(∘))
SubTerm
Subtract term
sin27^(∘) = cos63^(∘)
Similarly, cos 78^(∘) can be written in terms of sine.
cos θ = sin(90^(∘) -θ)
Substitute
θ= 78^(∘)
cos 78^(∘) = sin(90^(∘)- 78^(∘))
SubTerm
Subtract term
cos 78^(∘) = sin 12^(∘)
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Pop Quiz
Practice Converting Between Sine and Cosine
Determine the value of x that makes the equation true.
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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 x^(∘) be the measure of the angle and writes the following equation. sin x^(∘) = cos x^(∘) 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 x.
b How can the sine and cosine of complementary angles be used to determine the value of x?
Answer
a Type of Right Triangle: Isosceles Right Triangle
Value of x: 45
Value of x: 45
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 x^(∘) is 90^(∘)-x^(∘).
Solution
a Draw a right triangle with an acute angle that measures x^(∘).
The sine and cosine ratios of ∠ A can be written as follows. sin x^(∘) = BC/AC cos x^(∘) = AB/AC Since sin x^(∘) = cos x^(∘), the above values can be substituted into this equation.
sin x^(∘) = cos x^(∘)
SubstituteII
sin x^(∘)= BC/AC, cos x^(∘)= AB/AC
BC/AC= AB/AC
MultEqn
LHS * AC=RHS* AC
BC=AB
It has been found that BC= AB. Therefore, △ ABC is an isosceles triangle. By the Isosceles Triangle Theorem, m ∠ A and m∠ C are equal to each other.
Since the sum of the interior angles of △ ABC is 180^(∘), the value of x, can be found.
m∠ A + m∠ B + m∠ C= 180^(∘)
SubstituteValues
Substitute values
x^(∘)+ 90^(∘) + x^(∘) = 180^(∘)
x^(∘) = 45^(∘)
This means that the value of x is 45.
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 x^(∘) is 90^(∘)-x^(∘), the equation below holds true.
cos x^(∘) = sin(90^(∘)-x^(∘)) It is given that sin x^(∘) = cos x^(∘). By substituting sin(90^(∘)-x^(∘)) for cos x^(∘), the value of x can be found. sin x^(∘) = cos x^(∘) ⇕ sin x^(∘) = sin(90^(∘)-x^(∘)) For acute angles, the equation above is true when x^(∘) is equal to 90^(∘)-x^(∘).
Therefore, the value of x is 45.
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Example
Solving Problems Using the Relationship Between Sine and Cosine
Let ∠ A and ∠ B be two acute angles of a right triangle. The sine of ∠ A and the cosine of ∠ B are expressed as follows. ∙ & sin A = 3/17x-1/17 [1.3em] ∙ & cos B = 2/17x+2/17 What is the value of x?
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Hint
Complementary angles add up to 90^(∘).
Solution
Notice that ∠ A and ∠ B are complementary angles because they are acute angles of a right triangle.
m∠ A + m∠ B + 90^(∘) = 180^(∘) ⇕ m∠ B = 90^(∘)- m∠ A
Using the relationship between sine and cosine of complementary angles, the following equation can be written.
sin A = cos( 90^(∘) -A) ⇓ sin A = cos B
Finally, in the above equation, the given expressions for sin A and cos B can be substituted. By doing so, the value of x can be found.
sin A = cos B
SubstituteExpressions
Substitute expressions
3/17x-1/17 = 2/17x+2/17
▼
Solve for x
SubEqn
LHS-2/17x=RHS-2/17x
3/17x-1/17-2/17x = 2/17
CommutativePropAdd
Commutative Property of Addition
3/17x-2/17x-1/17 = 2/17
FactorOut
Factor out x
(3/17-2/17)x-1/17 = 2/17
SubFrac
Subtract fractions
1/17x-1/17 = 2/17
AddEqn
LHS+1/17=RHS+1/17
1/17x = 2/17+1/17
AddFrac
Add fractions
1/17x = 3/17
MoveRightFacToNumOne
1/b* a = a/b
x/17 = 3/17
MultEqn
LHS * 17=RHS* 17
x = 3
The value of x is 3.
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Example
Using the Relationship Between Sine and Cosine to Solve Problems
In a right triangle, an acute angle measures w and satisfies the following equation. sin(w/4+20) = cos w Find the value of w.
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Hint
Solution
Recall that the cosine of an acute angle is equal to the sine of its complementary angle.
cos θ = sin(90^(∘) -θ)
Since the complement of w is (90^(∘)-w), cos w is equal to sin(90^(∘)-w).
sin(w/4+20) = cos(w) ⇕ sin(w/4+20) = sin(90-w)
Because w is an acute angle, then ( w4+20) and (90-w) are also acute angles. Since the sines of these two angles are equal, the angles have the same measure.
w/4+20 = 90-w
w=56
The value of w is 56.
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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.
tan θ=cot(90^(∘) -θ) cot θ=tan(90^(∘) -θ)
Proof
Consider a right triangle with side lengths x, y, and z.
By using their definitions, the tangent and cotangent ratios can be written in terms of x and y. tan A = y/x & & cot A= x/y tan B= x/y & & cot B= y/x Since the acute angles of a right triangle are complementary, ∠ A and ∠ B are complementary angles. It can be seen that tan A=cot B and cot A=tan B. This is true for all pairs of complementary angles.
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Example
Solving Problems Using the Relationship Between Tangent and Cotangent
Find the value of the following expression. tan1^(∘) * tan2^(∘) * tan3^(∘) * ... * tan89^(∘)
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Hint
Solution
Consider the product of the first term and the last terms of the expression.
Since the complement of 89^(∘) is 1^(∘), by the tangent and cotangent relationship of complementary angles, it can be said that tan 89^(∘) is equal to cot 1^(∘).
Recall that the cotangent is the reciprocal of the tangent.
tan 1^(∘) * cot 1^(∘)
cot(θ) = 1/tan(θ)
tan 1^(∘) * 1/tan 1^(∘)
DenomMultFracToNumber
tan 1^(∘) * a/tan 1^(∘)= a
1
Therefore, the product of the first and the last terms is equal to 1. Similarly, the product of the second and second to last terms is equal to 1, and so on. The terms in this product can be grouped so that each pair has a product of 1. Since the given product has 89 terms, tan 45^(∘) will stand alone.
Note that tan 45^(∘) =1. Furthermore, the rest of the factors can be grouped by pairs such that their product is also 1. By the Identity Property of Multiplication, the value of the given expression is 1.
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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 30 feet below the surface of the ocean.
External credits: @brgfx
If cot α = 0.75, and cot β = 1.73, calculate the distance between the apparent depth and actual depth of the whale. Round the answer to the nearest foot.
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Hint
Solution
Begin by labeling points on the diagram.
External credits: @brgfx
Notice that B, D, and C are collinear. Therefore, the measures of ∠ FDE, ∠ FDA, and ∠ ADB add up to 180^(∘).
m∠ FDE + m∠ FDA+ m∠ ADB = 180^(∘)
SubstituteII
m∠ FDE= α, m∠ FDA= 90^(∘)
α + 90^(∘)+ m∠ ADB = 180^(∘)
m∠ ADB = 90^(∘)-α
This means that ∠ FDE and ∠ ADB are complementary angles. Consequently, the cotangent of α is equal to the tangent of its complementary angle. cot α = tan (90-α) From here, the distance between A and D can be calculated using tangent ratio of 90^(∘)-α. tan (90-α) = 30/AD ⇓ cot α = 30/AD Substitute the given value for cot α and solve for AD.
Now that AD is found, AC can be calculated. By the Alternate Interior Angles Theorem, β and ∠ ACD are congruent.
External credits: @brgfx
The cotangent ratio of β is then equal to AC40. Since cot β is known, AC can be found.
cot β = AC/40
Substitute
cot α= 1.73
1.73 = AC/40
▼
Solve for AC
MultEqn
LHS * 40=RHS* 40
69.2 = AC
RearrangeEqn
Rearrange equation
AC = 69.2
RoundInt
Round to nearest integer
AC ≈ 69
The whale is located at a depth of about 69 feet. Therefore, Ignacio who is located at E sees the whale about 39 feet closer to the surface. AC- AB & = 69 -30 & = 39 What a cool phenomenon made sense by complementary angles!
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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 co.
ccc
sine & cosine & tangent cosecant & secant & cotangent
Consider an example trigonometric equation.
sin α = cos β
In this case, the prefix co
denotes that β is the co-angle, or complementary angle, of α. The identities seen in this lesson are referred to as cofunction identities.
| Cofunction Identities | |
|---|---|
| sin θ = cos(90^(∘)-θ) | cos θ = sin(90^(∘)-θ) |
| tan θ = cot(90^(∘)-θ) | cot θ = tan(90^(∘)-θ) |
| sec θ = csc(90^(∘)-θ) | csc θ = sec(90^(∘)-θ) |
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Properties of the Trigonometric Ratios
Exercises
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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.
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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.
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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^(∘)
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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
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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.
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Properties of the Trigonometric Ratios
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