Math Problem Statement

Given the triangle in the figure, find the 6 trig ratios of θ. a = 3 and b = 4.

Solution

Let's solve this step by step.

Problem Summary:

We are tasked with finding the six trigonometric ratios for the angle θ\theta in the triangle, where:

  • a=3a = 3 (adjacent side to θ\theta),
  • b=4b = 4 (opposite side to θ\theta).

Step 1: Find the hypotenuse using the Pythagorean theorem:

c=a2+b2=32+42=9+16=25=5c = \sqrt{a^2 + b^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 So, the hypotenuse c=5c = 5.

Step 2: Write the trigonometric ratios:

  1. Sine (sin\sin): sinθ=oppositehypotenuse=bc=45\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{b}{c} = \frac{4}{5}

  2. Cosine (cos\cos): cosθ=adjacenthypotenuse=ac=35\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{a}{c} = \frac{3}{5}

  3. Tangent (tan\tan): tanθ=oppositeadjacent=ba=43\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{b}{a} = \frac{4}{3}

  4. Cosecant (csc\csc): cscθ=hypotenuseopposite=cb=54\csc \theta = \frac{\text{hypotenuse}}{\text{opposite}} = \frac{c}{b} = \frac{5}{4}

  5. Secant (sec\sec): secθ=hypotenuseadjacent=ca=53\sec \theta = \frac{\text{hypotenuse}}{\text{adjacent}} = \frac{c}{a} = \frac{5}{3}

  6. Cotangent (cot\cot): cotθ=adjacentopposite=ab=34\cot \theta = \frac{\text{adjacent}}{\text{opposite}} = \frac{a}{b} = \frac{3}{4}

Final Answer:

The six trigonometric ratios are: sinθ=45,cosθ=35,tanθ=43,cscθ=54,secθ=53,cotθ=34.\sin \theta = \frac{4}{5}, \quad \cos \theta = \frac{3}{5}, \quad \tan \theta = \frac{4}{3}, \quad \csc \theta = \frac{5}{4}, \quad \sec \theta = \frac{5}{3}, \quad \cot \theta = \frac{3}{4}.

Let me know if you'd like a more detailed breakdown!


Related Questions:

  1. What are the reciprocal relationships between trigonometric functions?
  2. How does the Pythagorean theorem relate to the unit circle?
  3. How do you find angles in degrees or radians from given trigonometric ratios?
  4. What happens to the trigonometric ratios if the triangle's sides are scaled up or down?
  5. How do you use these ratios in solving word problems or real-world scenarios?

Tip:

Memorizing the common right triangle ratios (e.g., 3-4-5, 5-12-13) can help speed up calculations without needing the Pythagorean theorem every time.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Right Triangles

Formulas

Pythagorean theorem: c = √(a^2 + b^2)
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
csc θ = hypotenuse / opposite
sec θ = hypotenuse / adjacent
cot θ = adjacent / opposite

Theorems

Pythagorean theorem

Suitable Grade Level

Grades 9-10