Textbook Question
Use the given information to find the exact value of each of the following: tan 2θ
sin θ = ﹣2/3, θ lies in quadrant III.
809
views
6. Trigonometric Identities and More Equations
Double Angle Identities
2:21 minutesProblem 18
Textbook Question
Write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. cos² 105° - sin² 105°
Verified step by step guidance1
Recognize that the expression \( \cos^2 105^\circ - \sin^2 105^\circ \) matches the form of the cosine double-angle identity: \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \).
Rewrite the given expression using the double-angle identity: \( \cos^2 105^\circ - \sin^2 105^\circ = \cos(2 \times 105^\circ) \).
Calculate the angle inside the cosine function: \( 2 \times 105^\circ = 210^\circ \), so the expression becomes \( \cos 210^\circ \).
Recall the exact value of \( \cos 210^\circ \) by considering its reference angle and quadrant. The reference angle is \( 210^\circ - 180^\circ = 30^\circ \), and since 210° is in the third quadrant where cosine is negative, \( \cos 210^\circ = -\cos 30^\circ \).
Use the exact value \( \cos 30^\circ = \frac{\sqrt{3}}{2} \) to write \( \cos 210^\circ = -\frac{\sqrt{3}}{2} \).
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Double-Angle Identities
Double-angle identities express trigonometric functions of twice an angle in terms of functions of the original angle. For cosine, the identity cos(2θ) = cos²θ - sin²θ directly relates to the given expression, allowing simplification by recognizing the pattern.
Recommended video:
05:06
Double Angle Identities
Exact Values of Trigonometric Functions
Exact values refer to precise trigonometric values for special angles, often expressed in terms of square roots and fractions. Knowing or deriving exact values for angles like 105° (which can be expressed as 60° + 45°) is essential for finding the exact value of the expression.
Recommended video:
6:04Introduction to Trigonometric Functions
Angle Sum and Difference Formulas
These formulas allow the calculation of sine, cosine, or tangent of sums or differences of angles. For example, cos(105°) can be found using cos(60° + 45°) = cos60°cos45° - sin60°sin45°, which helps in determining exact trigonometric values needed for the problem.
Recommended video:
2:25Verifying Identities with Sum and Difference Formulas
Related Videos
Related Practice