Sine Function

• 1). 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.
  
  
• 2). Complete the division on the right side of the equation.
  
  Sin(A) = 0.9
  
  
• 3). 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

• 1). 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.
  
  
• 2). Complete the division on the right side of the equation.
  
  Cos(A) = 0.75.
  
  
• 3). 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

• 1). 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.
  
  
• 2). Complete the division on the right side of the equation.
  
  Tan(B) = 0.4.
  
  
• 3). 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

• 1). 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.
  
  
• 2). Multiply both sides by the denominator of the fraction with the unknown angle.
  
  For instance, 6[sin(50)/5] = sin(B).
  
  
• 3). Simplify the expression.
  
  Sin(B) equals approximately 0.92.
  
  
• 4). Take the inverse sine of both sides to solve for the missing angle.
  
  Angle B is approximately 67 degrees.
  
  
• 5). 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.
  
  
• 6). 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

• 1). 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)).
  
  
• 2). Simplify the expression.
  
  The equation simplifies to 16 = 4 + 16 - 16cos(C) or 16 = 20 - 16cos(C).
  
  
• 3). Rearrange the equation algebraically to isolate cos(C).
  
  -0.25 = cos(C).
  
  
• 4). Take the inverse cosine of both sides of the equation to solve for C.
  
  Angle C equals approximately 104.5 degrees.