QuestionIn $\triangle Q R S$ , if $\sin R=\cos S$ , what can you conclude about $\angle R$ and $\angle S$ ?

Studdy Solution

STEP 1

Assumptions1. We are given a triangle QRS. . We know that $\sin R = \cos$ .

STEP 2

We know that the sine of an angle is equal to the cosine of its complement. This is a fundamental property of trigonometry.
$\sin \theta = \cos (90^\circ - \theta)$

STEP 3

Given that $\sin R = \cos$ , we can conclude that $=90^\circ - R$ .

STEP 4

Since we are dealing with a triangle, the sum of the angles in a triangle is $180^\circ$ . $R + + Q =180^\circ$

STEP 5

Substitute $=90^\circ - R$ into the equation.
$R + (90^\circ - R) + Q =180^\circ$

STEP 6

implify the equation.
$90^\circ + Q =180^\circ$

STEP 7

olve for $Q$ .
$Q =180^\circ -90^\circ =90^\circ$

STEP 8

Since $Q$ is $90^\circ$ , this implies that $R$ and are complementary angles because the sum of $R$ and must be $90^\circ$ to add up to $180^\circ$ with $Q$ .
So, the conclusion is that both $R$ and $$ are complementary angles.

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QuestionIn $\triangle Q R S$ , if $\sin R=\cos S$ , what can you conclude about $\angle R$ and $\angle S$ ?

Studdy Solution

STEP 1

Assumptions1. We are given a triangle QRS. . We know that $\sin R = \cos$ .

STEP 2

We know that the sine of an angle is equal to the cosine of its complement. This is a fundamental property of trigonometry.
$\sin \theta = \cos (90^\circ - \theta)$

STEP 3

Given that $\sin R = \cos$ , we can conclude that $=90^\circ - R$ .

STEP 4

Since we are dealing with a triangle, the sum of the angles in a triangle is $180^\circ$ . $R + + Q =180^\circ$

STEP 5

Substitute $=90^\circ - R$ into the equation.
$R + (90^\circ - R) + Q =180^\circ$

STEP 6

implify the equation.
$90^\circ + Q =180^\circ$

STEP 7

olve for $Q$ .
$Q =180^\circ -90^\circ =90^\circ$

STEP 8

Since $Q$ is $90^\circ$ , this implies that $R$ and are complementary angles because the sum of $R$ and must be $90^\circ$ to add up to $180^\circ$ with $Q$ .
So, the conclusion is that both $R$ and $$ are complementary angles.

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QuestionIn △QRS\triangle Q R S△QRS, if sin⁡R=cos⁡S\sin R=\cos SsinR=cosS, what can you conclude about ∠R\angle R∠R and ∠S\angle S∠S?

Studdy Solution

STEP 1

Assumptions1. We are given a triangle QRS. . We know that sin⁡R=cos⁡\sin R = \cossinR=cos.

STEP 2

We know that the sine of an angle is equal to the cosine of its complement. This is a fundamental property of trigonometry.sin⁡θ=cos⁡(90∘−θ)\sin \theta = \cos (90^\circ - \theta)sinθ=cos(90∘−θ)

STEP 3

Given that sin⁡R=cos⁡\sin R = \cossinR=cos, we can conclude that =90∘−R =90^\circ - R=90∘−R.

STEP 4

Since we are dealing with a triangle, the sum of the angles in a triangle is 180∘180^\circ180∘.R++Q=180∘R + + Q =180^\circR++Q=180∘

STEP 5

Substitute =90∘−R =90^\circ - R=90∘−R into the equation.R+(90∘−R)+Q=180∘R + (90^\circ - R) + Q =180^\circR+(90∘−R)+Q=180∘

STEP 6

implify the equation.90∘+Q=180∘90^\circ + Q =180^\circ90∘+Q=180∘

STEP 7

olve for QQQ.Q=180∘−90∘=90∘Q =180^\circ -90^\circ =90^\circQ=180∘−90∘=90∘

STEP 8

Since QQQ is 90∘90^\circ90∘, this implies that RRR and are complementary angles because the sum of $R$ and must be 90∘90^\circ90∘ to add up to 180∘180^\circ180∘ with QQQ.So, the conclusion is that both RRR and $$ are complementary angles.

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QuestionIn $\triangle Q R S$ , if $\sin R=\cos S$ , what can you conclude about $\angle R$ and $\angle S$ ?

Assumptions1. We are given a triangle QRS. . We know that $\sin R = \cos$ .

We know that the sine of an angle is equal to the cosine of its complement. This is a fundamental property of trigonometry.
$\sin \theta = \cos (90^\circ - \theta)$

Given that $\sin R = \cos$ , we can conclude that $=90^\circ - R$ .

Since we are dealing with a triangle, the sum of the angles in a triangle is $180^\circ$ . $R + + Q =180^\circ$

Substitute $=90^\circ - R$ into the equation.
$R + (90^\circ - R) + Q =180^\circ$

implify the equation.
$90^\circ + Q =180^\circ$

olve for $Q$ .
$Q =180^\circ -90^\circ =90^\circ$

Since $Q$ is $90^\circ$ , this implies that $R$ and are complementary angles because the sum of $R$ and must be $90^\circ$ to add up to $180^\circ$ with $Q$ .
So, the conclusion is that both $R$ and $$ are complementary angles.