/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 63 Using Cofunction Identities Use ... [FREE SOLUTION] | 91Ó°ÊÓ

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Using Cofunction Identities Use the cofunction identities to evaluate the expression without using a calculator. $$\sin ^{2} 25^{\circ}+\sin ^{2} 65^{\circ}$$

Short Answer

Expert verified
The expression \(\sin ^{2} 25^{\circ}+\sin ^{2} 65^{\circ}\) evaluates to 1.

Step by step solution

01

Modification of the given equation

Rewrite \(\sin ^{2} 65^{\circ}\) as \(\cos ^{2} 25^{\circ}\) using the cofunction identity \(\sin(90° - θ) = \cos(θ)\). This gives us:\(\sin ^{2} 25^{\circ}+\sin ^{2} 65^{\circ} = \sin ^{2} 25^{\circ} + \cos ^{2} 25^{\circ}\)
02

Apply the Pythagorean Trigonometric Identity

Using the Pythagorean Trigonometric Identity \(\sin^{2}θ + \cos^{2}θ = 1\), the equation simplifies to:\(\sin ^{2} 25^{\circ} + \cos ^{2} 25^{\circ} = 1\)
03

Final Answer

The final answer to the problem is therefore 1, as the statement \(\sin ^{2} 25^{\circ} + \cos ^{2} 25^{\circ} = 1\) holds true for any real value of θ.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Pythagorean Trigonometric Identity
The Pythagorean Trigonometric Identity is a fundamental concept in trigonometry. It states that for any angle \(\theta\), the square of the sine of \(\theta\) plus the square of the cosine of \(\theta\) equals 1.
This can be written as: \(\sin^2\theta + \cos^2\theta = 1\). This identity is derived from the Pythagorean Theorem and helps in simplifying expressions involving trigonometric functions.
  • This identity is useful when working with angles in both degrees and radians.
  • It applies to any angle, regardless of its size or quadrant.
In practical terms, if you know one of the trigonometric values (sine or cosine), you can easily find the other using this identity. This makes calculations involving angles seamless and straightforward.
Sine Function
The sine function is one of the basic trigonometric functions. It is defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle.
Mathematically, for an angle \(\theta\), it is expressed as:\[\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}}\]
  • The sine function is periodic with a period of \(360^{\circ}\) or \(2\pi\) radians.
  • Its value ranges from -1 to 1.
The function is essential in modeling wave patterns and oscillations. In our problem, the sine values of angles \(25^{\circ}\) and \(65^{\circ}\) interact with the cosine of their respective complementary angles due to trigonometric relationships like the cofunction identities.
Cosine Function
The cosine function, paired with the sine function, is another core trigonometric function. It is defined in the context of a right triangle as the ratio of the adjacent side to the hypotenuse.
The function is expressed as:\[\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}\]
  • Similar to sine, cosine also oscillates between -1 and 1.
  • It has a period of \(360^{\circ}\) or \(2\pi\) radians.
The cosine function is pivotal in various fields such as physics, engineering, and computer science for solving problems involving waves and cycles.
Additionally, by using the cofunction identity \(\sin(90^{\circ} - \theta) = \cos(\theta)\), we see how complementary angles relate and influence each other's sine and cosine values, allowing simplification of trigonometric expressions as shown in the problem given.

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Most popular questions from this chapter

The height \(h\) (in feet) above ground of a seat on a Ferris wheel at time \(t\) (in minutes) can be modeled by $$h(t)=53+50 \sin \left(\frac{\pi}{16} t-\frac{\pi}{2}\right)$$. The wheel makes one revolution every 32 seconds. The ride begins when \(t=0\). (a) During the first 32 seconds of the ride, when will a person on the Ferris wheel be 53 feet above ground? (b) When will a person be at the top of the Ferris wheel for the first time during the ride? If the ride lasts 160 seconds, then how many times will a person be at the top of the ride, and at what times?

Find all solutions of the equation in the interval \([0,2 \pi)\).$$\cos \left(x+\frac{\pi}{4}\right)+\cos \left(x-\frac{\pi}{4}\right)=1$$

Determine whether the statement is true or false. Justify your answer.$$\sin \left(x-\frac{\pi}{2}\right)=-\cos x$$

(a) use a graphing utility to graph the function and approximate the maximum and minimum points on the graph in the interval \(\mathbf{0}, \mathbf{2} \boldsymbol{\pi}\) ), and (b) solve the trigonometric equation and demonstrate that its solutions are the \(x\) -coordinates of the maximum and minimum points of \(f .\) (Calculus is required to find the trigonometric equation.) $$\begin{array}{cc}\text{Function} && \text {Trigonometric Equation} \\\ f(x)=2 \sin x+\cos 2 x && 2 \cos x-4 \sin x \cos x=0 \end{array}$$

Solving a Trigonometric Equation In Exercises \(69-74,\) find all solutions of the equation in the interval \([0,2 \pi)\).$$\cos \left(x+\frac{\pi}{4}\right)-\cos \left(x-\frac{\pi}{4}\right)=1$$

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