Ch. 3 - Trigonometric Identities and Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 3 - Trigonometric Identities and Equations

Problem 7c

Chapter 3, Problem 7c

In Exercises 7–14, use the given information to find the exact value of each of the following: c. tan 2θ 15 sin θ = -------- , θ lies in quadrant II. 17

Verified step by step guidance

Identify the given information: $\sin \theta = \frac{15}{17}$ and $\theta$ lies in quadrant II.

Recall that in quadrant II, sine is positive and cosine is negative. Use the Pythagorean identity to find $\cos \theta$: $\cos \theta = -\sqrt{1 - \sin^2 \theta}$.

Calculate $\cos \theta$ by substituting $\sin \theta = \frac{15}{17}$ into the identity: $\cos \theta = -\sqrt{1 - \left(\frac{15}{17}\right)^2}$.

Use the double-angle formula for tangent: $\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}$. To apply this, first find $\tan \theta = \frac{\sin \theta}{\cos \theta}$.

Substitute $\tan \theta$ into the double-angle formula to express $\tan 2\theta$ in terms of known values, then simplify the expression to find the exact value.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Ratios and Quadrants

Trigonometric ratios like sine, cosine, and tangent relate the angles of a triangle to the ratios of its sides. Knowing the quadrant of the angle is crucial because it determines the sign (positive or negative) of these ratios. For example, in quadrant II, sine is positive while cosine and tangent are negative.

Double-Angle Identity for Tangent

The double-angle identity for tangent states that tan(2θ) = (2 tan θ) / (1 - tan² θ). This formula allows you to find the tangent of twice an angle using the tangent of the original angle, which can be derived from sine and cosine values.

Finding Cosine and Tangent from Sine

Given sin θ and the quadrant, you can find cos θ using the Pythagorean identity cos² θ = 1 - sin² θ, adjusting the sign based on the quadrant. Then, tan θ is found by dividing sin θ by cos θ. These values are essential to apply the double-angle formula for tangent.

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Related Practice

Textbook Question

Use the given information to find the exact value of each of the following: $sine 2 theta$

$\sin θ = \frac{12}{13},$

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Textbook Question

Use the given information to find the exact value of each of the following: $tangent 2 theta$

$\sin θ = \frac{12}{13},$

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Textbook Question

Each expression is the right side of the formula for cos (α - β) with particular values for α and β. Write the expression as the cosine of an angle.

$\cos \frac{5 π}{12} \cos \frac{π}{12} + \sin \frac{5 π}{12} \sin \frac{π}{12}$

Textbook Question

Each expression is the right side of the formula for cos (α - β) with particular values for α and β. Find the exact value of the expression.

$\cos \frac{5 π}{12} \cos \frac{π}{12} + \sin \frac{5 π}{12} \sin \frac{π}{12}$

Textbook Question

Be sure that you've familiarized yourself with the first set of formulas presented in this section by working C1–C4 in the Concept and Vocabulary Check. In Exercises 1–8, use the appropriate formula to express each product as a sum or difference.

$\cos \frac{3 x}{2} \sin \frac{x}{2}$

Textbook Question

Use the given information to find the exact value of each of the following: $cosine 2 theta$

$\sin θ = \frac{12}{13},$