By Bryan Grubbs | Updated Aug 30, 2022
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In mathematics, the study of triangles is called trigonometry. By applying the core trigonometric functions—sine, cosine, and tangent—you can uncover unknown angles and side lengths. The unknown angle is commonly denoted as θ (theta). This guide explains how to calculate θ using right‑triangle shortcuts, the Law of Sines, and the Law of Cosines.
When a triangle contains a 90° angle, it is a right‑triangle. For these triangles, the well‑known mnemonic SOH‑CAH‑TOA helps you relate sides to angles:
To solve for θ, use the inverse trigonometric functions on a graphing calculator: arcsin (SIN−1), arccos (COS−1), and arctan (TAN−1). Input the side ratio in fractional form and ensure the calculator is in DEGREE mode.
Example: If the side opposite θ is 4 and the hypotenuse is 5, enter SIN−1(4/5). The result is approximately 53.13°.
For triangles that do not contain a right angle, the Law of Sines is your tool of choice when you know one angle and its opposite side. The relationship is:
sin A / a = sin B / b = sin C / c
To find an unknown angle, isolate its sine by multiplying both sides by the opposite side’s length. Then use the arcsine function.
Example: With side a = 5, side b = 7, and angle A = 45°, compute SIN−1((7 × SIN(45°))/5). The result is about 81.87°.
The Law of Cosines applies to any triangle and is especially useful when all three sides are known. The formula is:
c² = a² + b² – 2ab cos(C) → cos(C) = (a² + b² – c²) / (2ab)
Example: For sides 5, 7, and 10, enter COS−1((5² + 7² – 10²) / (2 × 5 × 7)). The calculator yields approximately 111.80°.
Remember that all triangles sum to 180°. By repeatedly applying these techniques to a variety of triangles, you’ll gain confidence and intuition for solving for θ. Mastery comes from practice and experimentation—each problem is a chance to refine your skill set.
