Math Problem Statement

Triangle A B C is shown. Angle A C B is a right angle. The length of hypotenuse A B is 12 centimeters, the length of C B is 9.8 centimeters, and the length of A C is 6.9 centimeters. Which expressions can be used to find m∠BAC? Select three options.

cos−1(StartFraction 6.9 Over 12 EndFraction) cos−1(StartFraction 9.8 Over 12 EndFraction) sin−1(StartFraction 6.9 Over 12 EndFraction) sin−1(StartFraction 9.8 Over 12 EndFraction) tan−1(StartFraction 6.9 Over 9.8 EndFraction)

Solution

Given the right triangle ABCABC with angle ACB\angle ACB as the right angle, we are asked to find expressions that can be used to find mBACm\angle BAC.

Let's identify the sides relative to BAC\angle BAC:

  • Opposite side: BC=9.8BC = 9.8 cm
  • Adjacent side: AC=6.9AC = 6.9 cm
  • Hypotenuse: AB=12AB = 12 cm

Now, let's analyze the trigonometric functions:

  1. Cosine function:
    cos(BAC)=AdjacentHypotenuse=ACAB=6.912\cos(\angle BAC) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{AC}{AB} = \frac{6.9}{12} So, mBAC=cos1(6.912)m\angle BAC = \cos^{-1}\left(\frac{6.9}{12}\right).

  2. Sine function:
    sin(BAC)=OppositeHypotenuse=BCAB=9.812\sin(\angle BAC) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{BC}{AB} = \frac{9.8}{12} So, mBAC=sin1(9.812)m\angle BAC = \sin^{-1}\left(\frac{9.8}{12}\right).

  3. Tangent function:
    tan(BAC)=OppositeAdjacent=BCAC=9.86.9\tan(\angle BAC) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{BC}{AC} = \frac{9.8}{6.9} To find mBACm\angle BAC, we use tan1(9.86.9)\tan^{-1}\left(\frac{9.8}{6.9}\right). However, the expression provided is tan1(6.99.8)\tan^{-1}\left(\frac{6.9}{9.8}\right), which would correspond to the complementary angle ABC\angle ABC, not BAC\angle BAC.

Given the options, the three correct expressions to find mBACm\angle BAC are:

  • cos1(6.912)\cos^{-1}\left(\frac{6.9}{12}\right)
  • sin1(9.812)\sin^{-1}\left(\frac{9.8}{12}\right)
  • None of the tangent-related expressions directly find mBACm\angle BAC.

Thus, the three correct options are:

  1. cos1(6.912)\cos^{-1}\left(\frac{6.9}{12}\right)
  2. sin1(9.812)\sin^{-1}\left(\frac{9.8}{12}\right)
  3. cos1(9.812)\cos^{-1}\left(\frac{9.8}{12}\right)

The third one actually is incorrect upon closer review (my mistake). So the accurate list includes the first and second plus the last option about tangent.

If you want more details or have any questions, feel free to ask! Here are some related questions:

  1. How do you calculate the sine, cosine, and tangent of an angle in a right triangle?
  2. What is the complementary angle, and how does it relate to trigonometric functions?
  3. How can the Pythagorean theorem help in verifying the side lengths in this triangle?
  4. What is the process to convert between degrees and radians in angle measurement?
  5. How do inverse trigonometric functions work, and when are they used?

Tip: Remember, when solving for an angle using trigonometric ratios, always ensure you're using the correct ratio (sine, cosine, or tangent) relative to the given sides of the triangle.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Right Triangle

Formulas

Trigonometric ratios: sine, cosine, tangent

Theorems

Pythagorean theorem

Suitable Grade Level

High School