Half-angle formulas are a set of trigonometric identities that allow for the simplification of expressions involving half-angles, such as $\sin(\theta/2)$ and $\cos(\theta/2)$. These formulas are particularly useful in the context of trigonometric integrals, as they can help reduce the complexity of the integrand.
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The half-angle formulas for sine and cosine are: $\sin(\theta/2) = \pm \sqrt{(1 - \cos(\theta))/2}$ and $\cos(\theta/2) = \pm \sqrt{(1 + \cos(\theta))/2}$.
The half-angle formula for tangent is: $\tan(\theta/2) = \frac{\sin(\theta)}{1 + \cos(\theta)}$.
Half-angle formulas are particularly useful when dealing with integrals involving trigonometric functions, as they can help simplify the integrand and facilitate the integration process.
The choice of the positive or negative sign in the sine and cosine half-angle formulas depends on the quadrant in which the half-angle lies.
Half-angle formulas can be derived from the angle addition and subtraction formulas, which are fundamental trigonometric identities.
Review Questions
Explain how the half-angle formulas can be used to simplify trigonometric integrals.
The half-angle formulas can be used to simplify trigonometric integrals by rewriting expressions involving half-angles, such as $\sin(\theta/2)$ and $\cos(\theta/2)$, in terms of the full-angle trigonometric functions. This can help reduce the complexity of the integrand, making the integration process more manageable. For example, substituting the half-angle formulas into an integral like $\int \sin(\theta/2) d\theta$ can transform it into an integral involving only full-angle trigonometric functions, which may be easier to evaluate.
Describe the relationship between the half-angle formulas and the angle addition and subtraction formulas.
The half-angle formulas can be derived from the angle addition and subtraction formulas, which are fundamental trigonometric identities. Specifically, the half-angle formulas can be obtained by setting one of the angles in the addition or subtraction formulas equal to half of the other angle. This relationship allows for the half-angle formulas to be expressed in terms of the full-angle trigonometric functions, providing a useful tool for simplifying expressions involving half-angles.
Analyze the importance of the choice of sign in the half-angle formulas for sine and cosine, and explain how it relates to the quadrant of the half-angle.
The choice of the positive or negative sign in the half-angle formulas for sine and cosine is crucial, as it depends on the quadrant in which the half-angle lies. The sign choice ensures that the resulting value of the sine or cosine function is consistent with the appropriate quadrant. For example, if the half-angle is in the first or second quadrant, the positive sign is used, while the negative sign is used for half-angles in the third or fourth quadrant. Properly accounting for the quadrant is important for ensuring the correct evaluation of the trigonometric expressions and their subsequent use in trigonometric integrals.
Trigonometric identities are equations that hold true for all values of the variables involved, allowing for the simplification and transformation of trigonometric expressions.
Angle Subtraction Formulas: Angle subtraction formulas are another set of trigonometric identities that express the sine, cosine, and tangent of the difference between two angles in terms of the sine, cosine, and tangent of the individual angles.
Trigonometric integrals are integrals that involve trigonometric functions, such as $\int \sin(x) dx$ or $\int \cos^2(x) dx$. The half-angle formulas can be used to simplify the integrand and facilitate the integration process.