By using Table 1, find the exact values of the following: i cos(45^∘) sin(45^∘) ii tan(60^∘)+tan

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By using Table 1, find the exact values of the following: i...

By using Table 1, find the exact values of the following: i $\cos \left(45^{\circ}\right) \sin \left(45^{\circ}\right)$ ii $\tan \left(60^{\circ}\right)+\tan \left(30^{\circ}\right)$ iii $\sin \left(60^{\circ}\right)+\tan \left(60^{\circ}\right)$ iv $\frac{\sin \left(45^{\circ}\right)}{\tan \left(45^{\circ}\right) \cos \left(45^{\circ}\right)}$ v $\cos \left(45^{\circ}\right)+\sin \left(60^{\circ}\right)$ vi $\sin \left(60^{\circ}\right) \cos \left(30^{\circ}\right)+\cos \left(60^{\circ}\right) \sin \left(30^{\circ}\right)$

By using Table 1, find the exact values of the following:
i $\cos \left(45^{\circ}\right) \sin \left(45^{\circ}\right)$
ii $\tan \left(60^{\circ}\right)+\tan \left(30^{\circ}\right)$
iii $\sin \left(60^{\circ}\right)+\tan \left(60^{\circ}\right)$
iv $\frac{\sin \left(45^{\circ}\right)}{\tan \left(45^{\circ}\right) \cos \left(45^{\circ}\right)}$
v $\cos \left(45^{\circ}\right)+\sin \left(60^{\circ}\right)$
vi $\sin \left(60^{\circ}\right) \cos \left(30^{\circ}\right)+\cos \left(60^{\circ}\right) \sin \left(30^{\circ}\right)$

Key Concepts

Trigonometric Identities

Trigonometric identities, such as the relationships between sine, cosine, and tangent, and the various formulae (e.g., product-to-sum or sum-to-product identities), are tools that allow one to simplify and manipulate expressions. These identities are vital in proving equalities and transforming complicated expressions into more manageable forms.

Angle Addition Formulas

The angle addition formulas, such as sin(A + B) = sin A cos B + cos A sin B, provide a method to compute trigonometric functions of sums of angles. They are especially useful when dealing with expressions that involve the sum of products of sine and cosine functions, allowing for the consolidation of terms into a single trigonometric function.

Special Angles

In trigonometry, special angles such as 30°, 45°, and 60° have exact trigonometric values that are commonly memorized. These known values simplify the evaluation of trigonometric functions and are essential when dealing with problems that require precise computations without approximations.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent describe relationships between angles and side lengths in right triangles, and they are also defined in the unit circle framework. Understanding these functions and how they interrelate—for example, knowing that tangent is the ratio of sine to cosine—is fundamental for solving trigonometric problems.

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