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Difficulty: HardRight Triangle Trigonometry (SOHCAHTOA)

In right triangle PQRPQR, the right angle is at vertex QQ. Point SS lies on leg PQPQ such that the length of PSPS is 77 units and the length of SQSQ is 55 units. If tan⁡(∠PRQ)=43\tan(\angle PRQ) = \frac{4}{3}, what is the value of cos⁡(∠SRQ)\cos(\angle SRQ)?

  1. A
    5106106\frac{5\sqrt{106}}{106}
  2. 9106106\frac{9\sqrt{106}}{106}Answer
  3. C
    9130130\frac{9\sqrt{130}}{130}
  4. D
    16281281\frac{16\sqrt{281}}{281}
  5. E
    59\frac{5}{9}

Answer

The value of cos⁡(∠SRQ)\cos(\angle SRQ) is 9106106\frac{9\sqrt{106}}{106}.
The total length of leg PQPQ is 7+5=127 + 5 = 12. Using the tangent definition in right triangle PQRPQR, tan⁡(∠PRQ)=PQQR=12QR=43\tan(\angle PRQ) = \frac{PQ}{QR} = \frac{12}{QR} = \frac{4}{3}, which gives QR=9QR = 9. Next, considering right triangle SQRSQR with legs SQ=5SQ = 5 and QR=9QR = 9, the hypotenuse SR=52+92=106SR = \sqrt{5^2 + 9^2} = \sqrt{106}. Finally, cos⁡(∠SRQ)=adjacenthypotenuse=QRSR=9106=9106106\cos(\angle SRQ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{QR}{SR} = \frac{9}{\sqrt{106}} = \frac{9\sqrt{106}}{106}.

Step-by-Step Solution

1
Find the length of leg PQPQ
PQ=PS+SQ=7+5=12PQ = PS + SQ = 7 + 5 = 12 units
Point SS lies on segment PQPQ, so the total length is the sum of its parts.
2
Calculate the length of leg QRQR using tan⁡(∠PRQ)\tan(\angle PRQ)
tan⁡(∠PRQ)=PQQR  ⟹  43=12QR  ⟹  QR=9\tan(\angle PRQ) = \frac{PQ}{QR} \implies \frac{4}{3} = \frac{12}{QR} \implies QR = 9 units
In right triangle PQRPQR, tangent is opposite side over adjacent side relative to ∠PRQ\angle PRQ.
3
Find hypotenuse SRSR of right triangle SQRSQR
SR=SQ2+QR2=52+92=25+81=106SR = \sqrt{SQ^2 + QR^2} = \sqrt{5^2 + 9^2} = \sqrt{25 + 81} = \sqrt{106} units
Apply the Pythagorean theorem to right triangle SQRSQR with right angle at QQ.
4
Determine cos⁡(∠SRQ)\cos(\angle SRQ) and rationalize the denominator
cos⁡(∠SRQ)=QRSR=9106=9106106\cos(\angle SRQ) = \frac{QR}{SR} = \frac{9}{\sqrt{106}} = \frac{9\sqrt{106}}{106}
Cosine is defined as the ratio of adjacent side over hypotenuse in right triangle SQRSQR.

Key Concept

Applying SOHCAHTOA and the Pythagorean Theorem in composite right triangle figures
Estimated Time:2m 0s
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Topics

Right Triangle Trigonometry (SOHCAHTOA)

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