Math Problem Statement

Decide if the given triangle is a right triangle. If it is, find the sine, cosine, and tangent for the angle B.

Solution

Let's analyze the image you provided.

Step 1: Checking if the Triangle is a Right Triangle

The triangle has sides labeled 3, 4, and 5. To determine if this is a right triangle, we can use the Pythagorean Theorem:

a2+b2=c2a^2 + b^2 = c^2

Here:

  • a=3a = 3 (one leg),
  • b=4b = 4 (the other leg), and
  • c=5c = 5 (the hypotenuse).

Now, check if the equation holds:

32+42=523^2 + 4^2 = 5^2 9+16=259 + 16 = 25 25=2525 = 25

Since the Pythagorean Theorem holds, this is indeed a right triangle.

Step 2: Finding the Trigonometric Ratios

Since it's a right triangle, we can now calculate the sine, cosine, and tangent for angle BB, where BB is opposite the side of length 4.

Sine of BB

sin(B)=oppositehypotenuse=45\sin(B) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{4}{5}

Cosine of BB

cos(B)=adjacenthypotenuse=35\cos(B) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{3}{5}

Tangent of BB

tan(B)=oppositeadjacent=43\tan(B) = \frac{\text{opposite}}{\text{adjacent}} = \frac{4}{3}

Final Answers:

  • sin(B)=45\sin(B) = \frac{4}{5}
  • cos(B)=35\cos(B) = \frac{3}{5}
  • tan(B)=43\tan(B) = \frac{4}{3}

Would you like further details or clarification? Here are some related questions to deepen your understanding:

  1. How does the Pythagorean Theorem help confirm if a triangle is right-angled?
  2. What would the sine, cosine, and tangent be for angle CC?
  3. How can you derive the Pythagorean identity from sine and cosine?
  4. What are the general conditions for a triangle to be classified as a right triangle?
  5. How can you use trigonometric ratios to find unknown angles or sides in right triangles?

Tip: Always verify the hypotenuse is the longest side when using the Pythagorean Theorem.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Pythagorean Theorem

Formulas

Pythagorean Theorem: a^2 + b^2 = c^2
Sine: sin(B) = opposite / hypotenuse
Cosine: cos(B) = adjacent / hypotenuse
Tangent: tan(B) = opposite / adjacent

Theorems

Pythagorean Theorem

Suitable Grade Level

Grades 9-10