Domanda del libro di testo
In Exercises 25–30, use an identity to find the value of each expression. Do not use a calculator.
sin 37° csc 37°
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 1, Problema 27
If θ is an acute angle and sin θ = (2√7) / 7, use the identity sin²θ + cos²θ = 1 to find cos θ.
Guida verificata passo dopo passo1
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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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.
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Percorso guidato
6:25
Pythagorean Identities
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.
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4:22Domain and Range of Function Transformations
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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6:36Quadratic Formula
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Domanda del libro di testo
In Exercises 25–32, the unit circle has been divided into eight equal arcs, corresponding to t-values of
0, 𝜋, 𝜋, 3𝜋, 𝜋, 5𝜋, 3𝜋, 7𝜋, and 2𝜋.
4 2 4 4 2 4
a. Use the (x,y) coordinates in the figure to find the value of the trigonometric function.
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