Math Problem Statement

Tentukan perbandingan trigonometri di sudut R & P.

Solution

The problem in the image is asking for trigonometric ratios based on the triangle provided.

The triangle shows:

  • Angle PP
  • RQ=3RQ = 3 (adjacent to P\angle P)
  • PQ=4PQ = 4 (opposite to P\angle P)
  • The hypotenuse PRPR, calculated as qq.

To solve for qq (the hypotenuse), the Pythagorean theorem is applied:

q=p2+r2q = \sqrt{p^2 + r^2} Where p=3p = 3 and r=4r = 4. Substituting in these values:

q=42+32=16+9=25=5q = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5

So, the hypotenuse q=5q = 5.

Trigonometric Ratios for Angle PP:

  1. Sine (sinP\sin P): sinP=oppositehypotenuse=PQPR=45\sin P = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{PQ}{PR} = \frac{4}{5}

  2. Cosine (cosP\cos P): cosP=adjacenthypotenuse=RQPR=35\cos P = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{RQ}{PR} = \frac{3}{5}

  3. Tangent (tanP\tan P): tanP=oppositeadjacent=PQRQ=43\tan P = \frac{\text{opposite}}{\text{adjacent}} = \frac{PQ}{RQ} = \frac{4}{3}

Let me know if you would like further clarification, more details on any step, or additional related topics explained!

Here are 5 follow-up questions based on this:

  1. How do you find the trigonometric ratios for angle RR in this triangle?
  2. Can you explain why the Pythagorean theorem is valid for right-angled triangles?
  3. How would the trigonometric values change if PQ=5PQ = 5 and RQ=12RQ = 12?
  4. What is the significance of the sine and cosine values being fractions?
  5. How can you use trigonometric ratios to find the missing angle PP?

Tip:

Always ensure you remember the basic trigonometric identities: sine, cosine, and tangent are defined with respect to the opposite, adjacent, and hypotenuse sides of a right triangle.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Right Triangle Ratios
Pythagorean Theorem

Formulas

Pythagorean Theorem: q = √(p² + r²)
Sine: sin(θ) = opposite / hypotenuse
Cosine: cos(θ) = adjacent / hypotenuse
Tangent: tan(θ) = opposite / adjacent

Theorems

Pythagorean Theorem

Suitable Grade Level

Grades 8-10