Ch. 1 - Angles and the Trigonometric Functions

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

Blitzer 3rd Edition

Ch. 1 - Angles and the Trigonometric Functions

Problem 27

Chapter 1, Problem 27

If θ is an acute angle and sin θ = (2√7) / 7, use the identity sin²θ + cos²θ = 1 to find cos θ.

Verified step by step guidance

Recall the Pythagorean identity for sine and cosine: \(\sin^{2}\theta + \cos^{2}\theta = 1\).

Substitute the given value of \(\sin \theta = \frac{2\sqrt{7}}{7}\) into the identity: \(\left(\frac{2\sqrt{7}}{7}\right)^{2} + \cos^{2}\theta = 1\).

Calculate \(\sin^{2}\theta\) by squaring \(\frac{2\sqrt{7}}{7}\): \(\left(\frac{2\sqrt{7}}{7}\right)^{2} = \frac{4 \times 7}{49} = \frac{28}{49}\).

Rewrite the equation as \(\frac{28}{49} + \cos^{2}\theta = 1\) and solve for \(\cos^{2}\theta\) by subtracting \(\frac{28}{49}\) from both sides.

Since \(\theta\) is acute, take the positive square root of \(\cos^{2}\theta\) to find \(\cos \theta\).

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Key Concepts

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

Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²θ + cos²θ = 1. This fundamental relationship allows us to find one trigonometric function if the other is known, by rearranging the equation to solve for the unknown value.

Sine Function and Its Range

The sine of an acute angle θ (0° < θ < 90°) is positive and represents the ratio of the length of the side opposite θ to the hypotenuse in a right triangle. Knowing sin θ helps determine cos θ using the Pythagorean identity.

Sign of Cosine in the First Quadrant

Since θ is acute, it lies in the first quadrant where both sine and cosine values are positive. This information is crucial when taking the square root to find cos θ, ensuring the correct positive value is chosen.

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