Trigonometric Ratios of Acute Angles

Dominika is helping Tadeo understand trigonometric ratios. She drew three right triangles for him to write trigonometric ratios with respect to the acute angle θ. Help Tadeo grasp this topic by selecting the correct answers!

a

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b

{"type":"choice","form":{"alts":["cos \u03b8=6\/10","cos \u03b8=8\/10","cos \u03b8=8\/6"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}

c

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Solution

a The sine of ∠ θ is defined as the ratio of the length of the side opposite ∠ θ to the hypotenuse of the right triangle. Therefore, Tadeo needs to identify these two sides.

It can be seen above that the hypotenuse of the right triangle is 5 and that the length of the opposite side to ∠ θ is 3. With this information, the sine of ∠ θ can be written.

sin θ = 3/5

b The cosine of ∠ θ is the ratio of the length of the adjacent side to ∠ θ to the hypotenuse of the right triangle. Therefore, Tadeo needs to identify these two sides.

The hypotenuse of the right triangle is 10 and the length of the adjacent side to ∠ θ is 6. With this information, the cosine of ∠ θ can be written.

cos θ = 6/10

c The tangent of ∠ θ is the ratio of the length of the side opposite ∠ θ to the length of the side adjacent ∠ θ. Therefore, Tadeo needs to identify these two sides.

It can be seen above that the lengths of the opposite and adjacent sides to ∠ θ are 12 and 5, respectively. With this information, the tangent of ∠ θ can be written.

tan θ = 12/5

Despite the awesome explanations Dominika provided, Tadeo still does not get how to find trigonometric ratios. To help his friend, Dominika thought of one more exercise.

This time, Dominika drew one right triangle and stated its three side lengths. She also labeled one of the triangle's acute angles.

Tadeo has now been asked to find the sine, cosine, and tangent of ∠ θ. Help him find the answers!

Solution

To find the sine, cosine, and tangent of ∠ θ, the hypotenuse of the right triangle needs to be identified. The opposite and adjacent sides to the angle also need to be identified.

It can be seen above that the hypotenuse is 17, and that the lengths of the opposite and adjacent sides to ∠ θ are 8 and 15, respectively. This information can be substituted into the definitions for sine, cosine, and tangent.

Definition	Substitute
sin θ = Length of oppositeside to∠ θ/Hypotenuse	sin θ = 8/17
cos θ = Length of adjacentside to∠ θ/Hypotenuse	cos θ = 15/17
tan θ = Length of oppositeside to∠ θ/Length of adjacentside to∠ θ	tan θ = 8/15

Tadeo finally understands the topic! But wait, Dominika wants to level up and has let him know that trigonometric ratios can also be used to find missing side lengths of a right triangle. "Tell me more," Tadeo responds. An acute angle and the hypotenuse of a right triangle are given. To see whether Tadeo masters this topic, Dominika asked him to find the value of x, which is the length of the opposite side to the given angle.

Help Tadeo find the value of x. If necessary, round the answer to three significant figures.

Solution

The hypotenuse of the right triangle is given, and the length of the opposite side to the given angle is to be found.

The trigonometric ratio that relates these two sides and the acute angle is the sine ratio. sin θ = Length of opposite side to∠ θ/Hypotenuse ⇓ sin 60^(∘)=x/10 Finally, the equation can be solved for x.

sin 60^(∘)=x/10

▼

Solve for x

MultEqn

LHS * 10=RHS* 10

sin 60^(∘) * 10=x/10 * 10

FracMultDenomToNumber

a/10* 10 = a

sin 60^(∘) * 10=x

Multiply

Multiply

10sin 60^(∘)=x

RearrangeEqn

Rearrange equation

x=10sin 60^(∘)

To find the value of sin 60^(∘), a calculator can be used. First, it must be set to degree mode. This is done by pushing MODE and selecting Degree in the third row.

Next the value of sin 60^(∘) can be calculated by pushing SIN followed by the angle measure.

Now the value of x can be calculated.

x=10sin 60^(∘)

▼

UseCalc

Use a calculator

x=10(0.866025...)

Multiply

Multiply

x=8.660254...

RoundSigDig

Round to 3 significant digit(s)

x≈ 8.66

The property seen before can be used, among other things, to find the sine or cosine ratio of an acute angle in a right triangle.

Kriz and his friends plan to spend Saturday afternoon playing video games. To optimize the space, they decide to tidy up the basement to ensure the console, snacks, and beverages are placed in the form of a right triangle. Kriz decides to set the snacks and the beverages 3 and 5 meters away from the console, respectively.

Previously, it was said that apart from being useful to find side lengths of a right triangle, trigonometric ratios can also be used to find missing angle measures.

Before playing video games with his friends, Kriz wants to finish his math homework to have a care-free weekend. He wants to find the measure of an acute angle in three different right triangles. By using the corresponding trigonometric ratios, help Kriz find m∠ θ in each triangle. Round the answer to the nearest degree.

a

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b

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":""},"formTextBefore":"m∠ \u03b8 =","formTextAfter":"^(∘)","answer":{"text":["74"]}}

c

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Solution

a In the given diagram, it can be seen that the lengths of the opposite and adjacent sides to ∠ θ are 35 and 12, respectively. The trigonometric ratio that relates these two sides is the tangent ratio.

tan θ = length of oppositeside to∠ θ/length of adjacentside to∠ θ ⇓ tan θ = 35/12 To solve this equation, the inverse of the tangent function could be used. tan θ = 35/12 ⇕ m∠ θ =tan ^(- 1) 35/12 To find the value of tan ^(- 1) 3512, a calculator should to be used. First, the calculator must be set in degree mode. This is done by pushing MODE and selecting Degree in the third row.

Next the value of tan ^(- 1) 3512, can be calculated by pushing 2ND, followed by TAN, and 35/12.

Thereofre, m∠ θ ≈ 71^(∘).

b In the given diagram, it is shown that the length of the adjacent side to ∠ θ is 7 and that the hypotenuse of the right triangle is 25. The trigonometric ratio that relates these two sides is the cosine ratio.

cos θ = length of adjacentside to∠ θ/hypotenuse ⇓ cos θ = 7/25 To solve this equation, the inverse of the cosine function is needed. cos θ = 7/25 ⇕ m∠ θ =cos ^(- 1) 7/25 To find the value of cos ^(- 1) 725, a calculator should be used. Just like before, the calculator must be set in degree mode. This is done by pushing MODE and selecting Degree in the third row.

Next the value of cos ^(- 1) 725, is calculated by pushing 2ND, followed by COS, and 7/25.

It was found that m∠ θ ≈ 74^(∘).

c In the diagram, it can be seen that the hypotenuse of the right triangle is 29 and that the length of the opposite side to ∠ θ is 20. The trigonometric ratio that relates these two sides is the sine ratio.

sin θ = length of oppositeside to∠ θ/hypotenuse ⇓ sin θ = 20/29 To solve this equation, the inverse of the sine function can be used. sin θ = 20/29 ⇕ m∠ θ =sin ^(- 1) 20/29 To find the value of sin ^(- 1) 2029, a calculator should be used. Just like in Parts A and B, the calculator must be set in degree mode by pushing MODE and selecting Degree in the third row.

Next the value of sin ^(- 1) 2029, can be calculated by pushing 2ND, followed by SIN, and 20/29.

Thereofre, m∠ θ ≈ 44^(∘).

If the sine, cosine, and tangent ratios are known, then their reciprocals cosecant, secant, and cotangent can be calculated without too much effort.

LaShay is really good at her favorite subject, Geometry. She has been appointed by Jefferson High's principal to do some tutoring for some of her classmates after school. To do so, she drew a right triangle. She then asked her peers to find all six trigonometric ratios with respect to the marked angle θ.

Help LaShay's classmates find the trigonometric ratios!

Solution

The hypotenuse of the right triangle and the opposite and adjacent sides to ∠ θ will be identified.

It can be seen that the hypotenuse is 101 and the lengths of the opposite and adjacent sides to ∠ θ are 20 and 99, respectively. With this information, the sine, cosine, and tangent ratios can be found. sin θ &= 20/101 [0.8em] cos θ &= 99/101 [0.8em] tan θ &= 20/99 The reciprocals of the above ratios are the cosecant, secant, and cotangent of ∠ θ. sin θ = 20/101 & ⇒ csc θ = 101/20 [0.9em] cos θ = 99/101 & ⇒ sec θ = 101/99 [0.9em] tan θ = 20/99 & ⇒ cot θ = 99/20