5.5 Double-angle identities
Trigonometry is derived from two Greek words, trigonon and metron which translate to triangle and measure respectively. Trigonometry is a branch of mathematics that tackles topics about how triangles, most especially, right triangles. This includes measuring their sides, the angles and solving for the missing parts.

We have discussed this topic and other related topics such as trigonometric functions in previous grade levels and chapters. In this chapter, we will focus more on the 6 trigonometric identities. These identities are better known for the mnemonic often used in school, SOH-CAH-TOA which stands for sine-opposite-hypotenuse, cosine-adjacent-hypotenuse and tangent-opposite-adjacent. If you would want to review on these identities, you can check out some Trigonometry identities formula sheet.

These identities are derived from the Pythagorean Theorem. From previous chapters, we learned that this theorem is represented by the a2+b2=c2a^2 + b^2 = c^2. In the equation a and b are values for the two sides of the right triangle while c is the hypotenuse or the longest side of the right triangle.

In this chapter, aside from reviewing the 6 trigonometric identities, we will also learn about the other kinds of identities. This chapter has 4 parts. In the first segment, we will look at quotient identities and reciprocal identities. For the quotient identities, an examples would be tan u=sinucosuu = \frac{sin u}{cos u}, while for the reciprocal identities, the example would be tan u=1cotuu = \frac{1}{ cot u}. Take note, that u stands for the measure of the angle. In most cases, you will often find θ\theta instead.

For the second part, we will look at Pythagorean identities. There are only three of these, namely, sin2x+cos2x=1sin^2x + cos^2x = 1, 1+cot2x=csc2x1 +cot^2x = csc^2x and tan2x+1=sec2xtan^2x + 1 = sec^2x. For the third part, we will study the sum and difference identities such as sin (α+β)(\alpha + \beta) = sin (α)(\alpha) cos (β)(\beta) + cos (α)(\alpha) sin (β)(\beta). There are five more of these and we will look at them in this chapter.

For the last segment of this chapter, we look at the double angle identities like sin 2α\alpha = 2sin α\alpha cos α\alpha and cos2 α\alpha = cos2cos^2 α\alpha sin2sin^2 α\alpha.

After all of the discussion here, you can use the concepts to simplify trigonometric expressions and prove trigonometric identities.

Double-angle identities

In this lesson, we will learn how to make use of the double-angle identities, a.k.a. double-angle formulas to find the sine and cosine of a double angle. It’s hard to simplify complex trigonometric functions without these formulas.

Lessons

Notes:
Download the Trigonometry identities chart here

Formulas:
sin2θ=2sinxcosx \sin 2\theta = 2\sin x\cos x
cos2θ=cos2xsin2x \cos 2 \theta = {\cos ^2}x - {\sin^2}x
=2cos2x1 = \;2{ \cos ^2}x - 1
=12sin2x = \;1 - 2{\sin ^2}x
tan2θ=2tanx1tan2x \tan 2 \theta = {{2\tan x} \over {1 - \tan ^2}x}
  • 1.
    Express each of the following in terms of a single trigonometric function:
  • 2.
    Prove identities
    • a)
      sin2x \sin 2x
    • b)
      sec2x\sec 2x
    • c)
      tan2x\tan 2x
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Double-angle identities

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