Find exact values for sinθ, cosθ, and tanθusing the information given. cos(2 θ)=(120)/(169) ; 2

step 1 So in these problems, we're asked to find values of sine of theta, cosine of theta, and tangent

Find exact values for sinθ, cosθ, and tanθusing the...

Key Concepts

Quadrant Analysis

Quadrant analysis is the process of determining the sign (positive or negative) of trigonometric functions based on the angle's location in the coordinate plane. Since the signs of sine, cosine, and tangent vary by quadrant, this consideration is crucial when using half-angle or other trigonometric formulas that involve taking square roots, ensuring that the correct values are chosen based on the given quadrant information.

Pythagorean Identity

The Pythagorean identity, given as sin²? + cos²? = 1, is a key relationship in trigonometry. It is fundamental for finding one trigonometric function when the other is known, especially after applying double or half-angle formulas where square roots arise. This identity ensures consistency and correctness in the values obtained for sine and cosine.

Half-Angle Formulas

Half-angle formulas are used to express the trigonometric functions of half an angle in terms of the trigonometric functions of the original angle. They are particularly useful when you have information about an angle that is twice what you need, allowing you to derive the sine, cosine, and tangent of the original angle from data given for twice that angle.

Double Angle Formulas

Double angle formulas express trigonometric functions of twice an angle in terms of functions of the original angle. For instance, they allow you to write cos(2?) in terms of cos²? or sin²?. These formulas are a gateway for connecting the given information about a multiple angle to the functions of the single angle, making them essential for solving problems where you need to switch between functions of 2? and ?.

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