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Please help:
express each trigonometric ratio as a fraction in simplist form​

Please help: express each trigonometric ratio as a fraction in simplist form​-example-1

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Answer:

sin (Q) = 15/17

sin (R) = 8/17

cos (Q) = 8/17

cos (R) = 15/17

tan (Q) = 15/8

tan (R) = 8/15

Explanation:

Step 1: Find the length of side QR (i.e., the hypotenuse):

  • Because the sine and cosine ratios require us to use the hypotenuse, we first need to find it.
  • Since this is a right triangle, we can find the hypotenuse using the Pythagorean Theorem, which is given by:

a^2 + b^2 = c^2, where

  • a and b are the triangle's shortest sides called legs,
  • and c is the longest side called the hypotenuse.

Thus, we can plug in 16 and 30 for a and b to find x, the hypotenuse (aka the length of side QR):

16^2 + 30^2 = c^2

256 + 900 = c^2

1156 = c^2

√1156 = √1156

34 = c

Thus, the length of side QR (the hypotenuse) is 34 units.

Step 2: Find sin Q and sin R:

sin Q:

The sine ratio is given by sin (θ) = opposite / hypotenuse, where

  • θ is the reference angle.
  • When angle Q is the reference angle, SR is the opposite side and QR is the hypotenuse.

Thus, sin (Q) = 30/34. This simplifies to sin (Q) = 15/17.

sin R:

When angle R is the reference angle, QS is the opposite side and QR is the hypotenuse.

Thus sin (R) = 16/34. This simplifies to sin(R) = 8/17.

Step 3: Find cos Q and cos R:

The cosine ratio is given by:

cos (θ) = adjacent / hypotenuse, where

  • θ is the reference angle.

cos Q:

  • When angle Q is the reference angle, QS is the adjacent side and QR is the hypotenuse.

Thus cos (Q) = 16/34. This simplifies to cos (Q) = 8/17.

cos R:

  • When angle R is the reference angle, SR is the adjacent side and QR is the hypotenuse.

Thus, cos (R) = 30/34. This simplifies to cos (R) = 15/17.

Step 4: Find tan Q and tan R:

The tangent ratio is given by:

tan (θ) = opposite / adjacent, where

  • θ is the reference angle.

tan (Q):

  • When angle Q is the reference angle, SR is the opposite side and QS is the adjacent side.

Thus tan (Q) = 30/16. This simplifies to tan (Q) = 15/8.

tan (R):

  • When angle R is the reference angle, QS is the opposite side and SR is the adjacent side.

Thus, tan (R) = 16/30. This simplifies to tan (R) = 8/15.

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