Textbook Question
In Exercises 39–42, use double- and half-angle formulas to find the exact value of each expression.cos² 15° - sin² 15°
691
views
6. Trigonometric Identities and More Equations
Double Angle Identities
3:33 minutesProblem 53
Textbook Question
In Exercises 47–54, use the figures to find the exact value of each trigonometric function.θ θ2 sin ------- cos -------2 2
Verified step by step guidance1
Identify the trigonometric identity for \( \sin(2\theta) \) and \( \cos(2\theta) \).
Recall that \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \) and \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \).
Substitute the given values of \( \sin(\theta) \) and \( \cos(\theta) \) from the figure into the identities.
Calculate \( \sin(2\theta) \) using the identity \( 2\sin(\theta)\cos(\theta) \).
Calculate \( \cos(2\theta) \) using the identity \( \cos^2(\theta) - \sin^2(\theta) \).
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
3mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Functions
Trigonometric functions, such as sine (sin) and cosine (cos), relate the angles of a triangle to the ratios of its sides. For a given angle θ in a right triangle, sin(θ) is the ratio of the length of the opposite side to the hypotenuse, while cos(θ) is the ratio of the adjacent side to the hypotenuse. Understanding these functions is essential for solving problems involving angles and lengths.
Recommended video:
6:04
Introduction to Trigonometric Functions
Double Angle Formulas
Double angle formulas are trigonometric identities that express trigonometric functions of double angles (2θ) in terms of single angles (θ). For example, sin(2θ) = 2sin(θ)cos(θ) and cos(2θ) = cos²(θ) - sin²(θ). These formulas are crucial for simplifying expressions and calculating exact values of trigonometric functions when dealing with angles that are multiples of a given angle.
Recommended video:
05:06Double Angle Identities
Exact Values of Trigonometric Functions
Exact values of trigonometric functions refer to the specific values of sin, cos, and other functions at key angles, such as 0°, 30°, 45°, 60°, and 90°. These values can be derived from the unit circle or special triangles. Knowing these exact values allows for quick calculations and is fundamental in solving trigonometric equations and problems.
Recommended video:
6:04Introduction to Trigonometric Functions
5:06mWatch next
Master Double Angle Identities with a bite sized video explanation from Patrick
Start learningRelated Videos
Related Practice