If \theta is an acute angle of a right triangle and Cos \theta = \frac{3}{8}, what is...

Question:

If {eq}\theta {/eq} is an acute angle of a right triangle and {eq}Cos \theta = \frac{3}{8}, {/eq} what is the value of {eq}Cosec\theta {/eq}?

Trigonometric Ratios:

The trigonometric ratios relate the three sides of a right-angled triangle to its angles. The six basic trigonometric ratios are sine (sin), cosine (cos), tangent (tan), secant (sec) cosecant (csc) and cotangent (cot).

Answer and Explanation:


If angle {eq}\theta {/eq} is an acute angle in a right-angled triangle, then {eq}\cos theta {/eq} relates the angle to the adjacent side and the opposite side to the angle. It is equal to:

  • {eq}\cos \theta = \rm \dfrac{Adjacent }{Hypotenuse} {/eq}

Recall that {eq}\csc \theta = \dfrac{1}{\cos \theta} {/eq}. Therefore:

  • {eq}\csc \theta = \dfrac{1}{\cos \theta} = \rm \dfrac{Hypotenuse}{Adjacent } {/eq}

Therefore, if:

  • {eq}\cos \theta = \dfrac{3}{8} {/eq}, then:
  • {eq}\csc \theta = \dfrac{1}{\cos \theta} = \boxed{\dfrac{8}{3}} {/eq}

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Trigonometric Ratios and Similarity

from Geometry: High School

Chapter 14 / Lesson 7
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