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Explore

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

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

$sinθ$ | $cosθ$ | |
---|---|---|

$θ=15_{∘}$ | $0.258819…$ | $0.965925…$ |

$θ=30_{∘}$ | $0.5$ | $0.866025…$ |

$θ=45_{∘}$ | $0.707106…$ | $0.707106…$ |

$θ=60_{∘}$ | $0.866025…$ | $0.5$ |

$θ=75_{∘}$ | $0.965925…$ | $0.258819…$ |

$sin30_{∘}cos75_{∘} == cos60_{∘}sin15_{∘} == 0.50.258819… $

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.

$sinθ=cos(90_{∘}−θ)cosθ=sin(90_{∘}−θ) $

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.
$sinA=zy sinB=zx cosA=zx cosB=zy $

By the Transitive Property of Equality, it can be said that $sinA=cosB$ and that $cosA=sinB.$
$⎩⎪⎪⎪⎨⎪⎪⎪⎧ sinA=zy cosB=zy ⇒sinA=cosB⎩⎪⎪⎪⎨⎪⎪⎪⎧ cosA=zx sinB=zx ⇒cosA=sinB $

This is true for all pairs of complementary angles.

Example

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

### Solution

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

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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.

To write $sin27_{∘}$ 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 $90_{∘}.$
Similarly, $cos78_{∘}$ can be written in terms of sine.

$sinθ=cos(90_{∘}−θ)$

Substitute

$θ=27_{∘}$

$sin27_{∘}=cos(90_{∘}−27_{∘})$

SubTerm

Subtract term

$sin27_{∘}=cos63_{∘}$

$cosθ=sin(90_{∘}−θ)$

Substitute

$θ=78_{∘}$

$cos78_{∘}=sin(90_{∘}−78_{∘})$

SubTerm

Subtract term

$cos78_{∘}=sin12_{∘}$

Pop Quiz

Determine the value of $x$ that makes the equation true.

Example

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.
### Answer

### Hint

### Solution

The sine and cosine ratios of $∠A$ can be written as follows.
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.
This means that the value of $x$ is $45.$

$sinx_{∘}=cosx_{∘} $

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 $x?$

a **Type of Right Triangle:** Isosceles Right Triangle

**Value of **$x:$ $45$

b See solution.

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 $90_{∘}−x_{∘}.$

a Draw a right triangle with an acute angle that measures $x_{∘}.$

$sinx_{∘}=ACBC cosx_{∘}=ACAB $

Since $sinx_{∘}=cosx_{∘},$ the above values can be substituted into this equation.
$sinx_{∘}=cosx_{∘}$

SubstituteII

$sinx_{∘}=ACBC $, $cosx_{∘}=ACAB $

$ACBC =ACAB $

MultEqn

$LHS⋅AC=RHS⋅AC$

$BC=AB$

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.

$cosx_{∘}=sin(90_{∘}−x_{∘}) $

It is given that $sinx_{∘}=cosx_{∘}.$ By substituting $sin(90_{∘}−x_{∘})$ for $cosx_{∘},$ the value of $x$ can be found.
$sinx_{∘}=cosx_{∘}⇕sinx_{∘}=sin(90_{∘}−x_{∘}) $

For acute angles, the equation above is true when $x_{∘}$ is equal to $90_{∘}−x_{∘}.$
Therefore, the value of $x$ is $45.$ Example

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.
### Hint

### Solution

$∙∙ sinA=173 x−171 cosB=172 x+172 $

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Complementary angles add up to $90_{∘}.$

Notice that $∠A$ and $∠B$ are complementary angles because they are acute angles of a right triangle.
The value of $x$ is 3.

$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.
$sinA=cos(90_{∘}−A)⇓sinA=cosB $

Finally, in the above equation, the given expressions for $sinA$ and $cosB$ can be substituted. By doing so, the value of $x$ can be found.
$sinA=cosB$

SubstituteExpressions

Substitute expressions

$173 x−171 =172 x+172 $

Solve for $x$

SubEqn

$LHS−172 x=RHS−172 x$

$173 x−171 −172 x=172 $

CommutativePropAdd

Commutative Property of Addition

$173 x−172 x−171 =172 $

FactorOut

Factor out $x$

$(173 −172 )x−171 =172 $

SubFrac

Subtract fractions

$171 x−171 =172 $

AddEqn

$LHS+171 =RHS+171 $

$171 x=172 +171 $

AddFrac

Add fractions

$171 x=173 $

MoveRightFacToNumOne

$b1 ⋅a=ba $

$17x =173 $

MultEqn

$LHS⋅17=RHS⋅17$

$x=3$

Example

In a right triangle, an acute angle measures $w$ and satisfies the following equation.
### Hint

### Solution

$sin(4w +20)=cosw $

Find the value of $w.$ {"type":"choice","form":{"alts":["<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">5<\/span><span class=\"mord\">6<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">3<\/span><span class=\"mord\">4<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">4<\/span><span class=\"mord\">5<\/span><\/span><\/span><\/span>","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:2.00744em;vertical-align:-0.686em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">3<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">8<\/span><span class=\"mord\">0<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span>"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}

Recall that the cosine of an acute angle is equal to the sine of its complementary angle.
The value of $w$ is 56.

$cosθ=sin(90_{∘}−θ) $

Since the complement of $w$ is $(90_{∘}−w),$ $cosw$ is equal to $sin(90_{∘}−w).$
$sin(4w +20)=cos(w)⇕sin(4w +20)=sin(90−w) $

Because $w$ is an acute angle, then $(4w +20)$ and $(90−w)$ are also acute angles. Since the sines of these two angles are equal, the angles have the same measure.
$4w +20=90−w$

$w=56$

Discussion

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.

$tanθ=cot(90_{∘}−θ)cotθ=tan(90_{∘}−θ) $

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.$$tanA=xy tanB=yx cotA=yx cotB=xy $

Since the acute angles of a right triangle are complementary, $∠A$ and $∠B$ are complementary angles. It can be seen that $tanA=cotB$ and $cotA=tanB.$ This is true for all pairs of complementary angles.
Example

Find the value of the following expression.
### Hint

### Solution

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, $tan45_{∘}$ will stand alone.

$tan1_{∘}×tan2_{∘}×tan3_{∘}×⋯×tan89_{∘} $

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Use the tangent and cotangent relationship of complementary angles.

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 $tan89_{∘}$ is equal to $cot1_{∘}.$ Recall that the cotangent is the reciprocal of the tangent.$tan1_{∘}×cot1_{∘}$

$cot(θ)=tan(θ)1 $

$tan1_{∘}×tan1_{∘}1 $

DenomMultFracToNumber

$tan1_{∘}⋅tan1_{∘}a =a$

$1$

Note that $tan45_{∘}=1.$ 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 $1.$

Example

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

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Use the tangent and cotangent relationship of complementary angles.

Begin by labeling points on the diagram.

External credits: @brgfx

$m∠FDE+m∠FDA+m∠ADB=180_{∘}$

SubstituteII

$m∠FDE=α$, $m∠FDA=90_{∘}$

$α+90_{∘}+m∠ADB=180_{∘}$

$m∠ADB=90_{∘}−α$

$cotα=tan(90−α) $

From here, the distance between $A$ and $D$ can be calculated using tangent ratio of $90_{∘}−α.$ $tan(90−α)=AD30 ⇓cotα=AD30 $

Substitute the given value for $cotα$ 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

$cotβ=40AC $

Substitute

$cotα=1.73$

$1.73=40AC $

Solve for $AC$

MultEqn

$LHS⋅40=RHS⋅40$

$69.2=AC$

RearrangeEqn

Rearrange equation

$AC=69.2$

RoundInt

Round to nearest integer

$AC≈69$

$AC−AB =69−30=39 $

What a cool phenomenon made sense by complementary angles! Closure

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$co.$

$sinecosecant cosinesecant tangentcotangent $

Consider an example trigonometric equation.
$sinα=cosβ $

In this case, the prefix codenotes 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 | |
---|---|

$sinθ=cos(90_{∘}−θ)$ | $cosθ=sin(90_{∘}−θ)$ |

$tanθ=cot(90_{∘}−θ)$ | $cotθ=tan(90_{∘}−$ |