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{{ printedBook.courseTrack.name }} {{ printedBook.name }} 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.

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

$sinθ$ | $cosθ$ | |
---|---|---|

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

$θ=30_{∘}$ | $0.5$ | $0.866025…$ |

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

$θ=60_{∘}$ | $0.866025…$ | $0.5$ |

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

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 $15_{∘}.$ $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.

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

$cos78_{∘}=sin12_{∘}$

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

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. $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 $x_{∘}.$

$sinx_{∘}=cosx_{∘}$

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 $45.$

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. $∙∙ sinA=173 x−171 cosB=172 x+172 $ What is the value of $x?$

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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.
$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.
The value of $x$ is 3.

$sinA=cosB$

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

$LHS⋅17=RHS⋅17$

$x=3$

In a right triangle, an acute angle measures $w$ and satisfies the following equation.
$sin(4w +20)=cosw $
Find the value of $w.$ ### Hint

### Solution

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

$4w +20=90−w$

$w=56$

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.

Find the value of the following expression.
$tan1_{∘}×tan2_{∘}×tan3_{∘}×⋯×tan89_{∘} $ ### 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.

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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 $1.$

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.

If $cotα=0.75,$ and $cotβ=1.73,$ calculate the distance between the apparent depth and actual depth of the whale. Round the answer to the nearest foot.{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.77001953125em;vertical-align:-0.009765625em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Regular\">ft<\/span><\/span><\/span><\/span><\/span>","answer":{"text":["39"]}}

Use the tangent and cotangent relationship of complementary angles.

Begin by labeling points on the diagram.

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

$m∠FDE=α$, $m∠FDA=90_{∘}$

$α+90_{∘}+m∠ADB=180_{∘}$

$m∠ADB=90_{∘}−α$

$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

$AC≈69$

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 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 | |
---|---|

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

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

$secθ=csc(90_{∘}−θ)$ | $cscθ=sec(90_{∘}−θ)$ |

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