A radian is a numerical ratio for any central angle that compares the magnitude of the intercepted arc length to the length of the circle's radius. This tells us that when the central angle θ equals 1 radian, the arc length **s** equals the radius. The expression s = rθ represents this relationship.
Note that when the arc length equals an entire circumference,
s = rθ 2πr = rθ θ = 2π^{ radians} = 360º
When measured in radians, as θ → 0, tan θ → sin θ which in turn approaches θ. This is called the small angle approximation and incurs an error of no greater than 0.1% for angles less than 5º. You can verify these relationships by examining the values for θ, sin θ, tan θ in Table 2.
All of these angle values are often represented graphically when we speak of circular functions. Trigonometrically, we generally use the variable **x** when expressing angles in terms of radians and **θ** when expressing them in terms of degrees.
Another useful set of trig identities are the called the Pythagorean Identities. These identities are based on line values drawn to a unit circle.
The double-angle formulas for sine and cosine.
sin 2θ = 2 sin θ cos θ cos 2θ = cos 2θ - sin 2θ
When solving for a missing side or angle is a triangle, there are two important relationships that apply to any triangle that can make your job easier: the **Law of Sines** and the **Law of Cosines**.
The Law of Sines, , can be used to solve for a missing side or angle in a general triangle when you know either
- two sides and an angle opposite one side, or
- only one side and all of the angles
The Law of Cosines can be used to solve for a missing side in a general triangle when you know the other two sides and their included angle. |