How to Solve Triangles Using Sin, Cos & Tan
The functions sine, cosine and tangent describe the position of a right triangle's angles and the length of its sides. The functions compare an angles opposite side and adjacent side to the hypotenuse of a triangle. In a right triangle, the hypotenuse, the longest side of the triangle, sits opposite the right angle. Typically, sine, cosine and tangent problems provide you with limited information about the triangle, such as two angles or all three lengths but no angles.
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Instructions
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Draw a diagram of the triangle. Label the right angle as "C." Label the hypotenuse, the side of the triangle opposite the right angle, as "c." Label the top angle as "A" and the side of the triangle opposite the top angle as "a." Label the remaining angle "B" and the remaining side "b."
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Write all of the known information. For example, write C = 90 to explain that C contains the right angle. Typically, a problem provides you with an additional angle or information about the length of one or two sides of the triangle. For example, if you know that the hypotenuse is 13 inches long, write c = 13.
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Solve for the length of the triangle, if possible. Write the Pythagorean theorem, a^2 + b^2 = c^2 and plug in the available variables. Subtract or add the numbers and find the square root. For example, write a^2 + 10^2 = 13^2, if the length of b is 10. Solve the equation to find that a is the square root of 69.
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Use sine, tangent or cosine to solve for a triangle's length, when you know the angle. Write the formula for sine, sin(A) = a/c, the formula for cosine, cos(A) = b/c, or the formula for tangent, tan(A) = a/b. Plug in the available variables and simplify the equation. For example, write sin(60) = a/13. Simplify to find a = sin(60) x 13. Use your calculator to solve for the sine and multiply it by the value for c, in this case 13.
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Solve for a missing angle using sine, cosine or tangent. Write the appropriate formula and fill in the variables. For example, write cosine(A)= 10/13. Multiply each side of the equation by the inverse of the function, in this case cos^-1, to solve for A and find A = (10/13)(cos^-1). Use your calculator to combine the values.
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References
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