How to Find the Angles of a Triangle With Equations

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Often, when mathematicians and scientists use triangles, they are not drawing two-dimensional shapes that can be measured with a protractor. Instead, they construct theoretical triangles that must be solved using equations. There are many of these trigonometric equations, each appropriate in a different circumstance. Sine, cosine and tangent functions allow you to find the missing angles of a right triangle. The Laws of Sines and Cosines allow you to calculate the angles of any other triangle.

Things You'll Need

  • Scientific calculator

Sine Function

  • Substitute the value of the side opposite the unknown angle and the length of the hypotenuse into the sine function, sin(x) = opposite/hypotenuse.

    For example, sin(A) = 9/10.

  • Complete the division on the right side of the equation.

    Sin(A) = 0.9

  • Take the inverse sine, or arcsine, of both sides. The answer is the measurement of the angle.

    Angle A equals approximately 64 degrees.

Cosine Function

  • Substitute the value of the side adjacent to the unknown angle and the length of the hypotenuse into the cosine function, cos(x) = adjacent/hypotenuse.

    For example, cos(A) = 3/4.

  • Complete the division on the right side of the equation.

    Cos(A) = 0.75.

  • Take the inverse cosine, or arccosine, of both sides. The answer is the measurement of the angle.

    Angle A equals approximately 41 degrees.

Tangent Function

  • Substitute the value of the side opposite the unknown angle and the length of the side adjacent to it into the tangent function, tan(x) = opposite/adjacent.

    For example, tan(B) = 2/5.

  • Complete the division on the right side of the equation.

    Tan(B) = 0.4.

  • Take the inverse tangent, or arctangent, of both sides. The answer is the measurement of the angle.

    Angle B equals approximately 22 degrees.

Law of Sines

  • Substitute the values of two sides and one angle into the Law of Sines equation, sin(A)/a = sin(B)/b = sin(C)/c. You can select any two of the three ratios.

    For example, if a = 5 cm, b = 6 cm and A = 50 degrees, the equation would become sin(50)/5 = sin(B)/6.

  • Multiply both sides by the denominator of the fraction with the unknown angle.

    For instance, 6[sin(50)/5] = sin(B).

  • Simplify the expression.

    Sin(B) equals approximately 0.92.

  • Take the inverse sine of both sides to solve for the missing angle.

    Angle B is approximately 67 degrees.

  • Examine the values of the other angles to determine if the angle seems reasonable. If the angles do not add up to 180 degrees, this is a reference angle, not a final answer.

    For instance, if the other angles measure 56 and 57 degrees, 67 degrees is reasonable, since 56 + 57 + 67 = 180. But if the other angles are 33 and 34 degrees, then the triangle needs a larger angle to bring the total to 180.

  • Subtract the reference angle from 180 to find the measurement of the angle.

    180 - 67 = 113; therefore, angle B is approximately 113 degrees.

Law of Cosines

  • Substitute the lengths of the triangle's sides into the Law of Cosines equation, c^2 = a^2 + b^2 - 2ab(cos(C)).

    For instance, if a = 2 cm, b = 4 cm and c = 4 cm, then the equation would become 4^2 = 2^2 + 4^2 - 2(2)(4)(cos(C)).

  • Simplify the expression.

    The equation simplifies to 16 = 4 + 16 - 16cos(C) or 16 = 20 - 16cos(C).

  • Rearrange the equation algebraically to isolate cos(C).

    -0.25 = cos(C).

  • Take the inverse cosine of both sides of the equation to solve for C.

    Angle C equals approximately 104.5 degrees.

References

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