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Define the Pythagorean Identity.

The fundamental trigonometric identity: sin⁡2(x)+cos⁡2(x)=1\sin^2(x) + \cos^2(x) = 1.

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Define the Pythagorean Identity.

The fundamental trigonometric identity: sin⁡2(x)+cos⁡2(x)=1\sin^2(x) + \cos^2(x) = 1.

Define tangent in terms of sine and cosine.

tan⁡(x)=sin⁡(x)cos⁡(x)\tan(x) = \frac{\sin(x)}{\cos(x)}

Define secant in terms of cosine.

sec⁡(x)=1cos⁡(x)\sec(x) = \frac{1}{\cos(x)}

Define cotangent in terms of sine and cosine.

cot⁡(x)=cos⁡(x)sin⁡(x)\cot(x) = \frac{\cos(x)}{\sin(x)}

Define cosecant in terms of sine.

csc⁡(x)=1sin⁡(x)\csc(x) = \frac{1}{\sin(x)}

What is the Pythagorean identity relating tangent and secant?

1+tan⁡2(x)=sec⁡2(x)1 + \tan^2(x) = \sec^2(x)

What is the Pythagorean identity relating cotangent and cosecant?

1+cot⁡2(x)=csc⁡2(x)1 + \cot^2(x) = \csc^2(x)

What is the sine sum identity?

sin⁡(a+b)=sin⁡(a)cos⁡(b)+cos⁡(a)sin⁡(b)\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)

What is the cosine sum identity?

cos⁡(a+b)=cos⁡(a)cos⁡(b)−sin⁡(a)sin⁡(b)\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)

What is the sine difference identity?

sin⁡(a−b)=sin⁡(a)cos⁡(b)−cos⁡(a)sin⁡(b)\sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b)

What is the cosine difference identity?

cos⁡(a−b)=cos⁡(a)cos⁡(b)+sin⁡(a)sin⁡(b)\cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b)

What is the sine double-angle identity?

sin⁡(2a)=2sin⁡(a)cos⁡(a)\sin(2a) = 2\sin(a)\cos(a)

What is the cosine double-angle identity (first form)?

cos⁡(2a)=cos⁡2(a)−sin⁡2(a)\cos(2a) = \cos^2(a) - \sin^2(a)

What is the cosine double-angle identity (second form)?

cos⁡(2a)=1−2sin⁡2(a)\cos(2a) = 1 - 2\sin^2(a)

What is the cosine double-angle identity (third form)?

cos⁡(2a)=2cos⁡2(a)−1\cos(2a) = 2\cos^2(a) - 1

How do you simplify an expression like 2sin⁡2(x)+2cos⁡2(x)−12\sin^2(x) + 2\cos^2(x) - 1?

  1. Factor out the 2: 2(sin⁡2(x)+cos⁡2(x))−12(\sin^2(x) + \cos^2(x)) - 1. 2. Apply the Pythagorean identity: 2(1)−12(1) - 1. 3. Simplify: 2−1=12 - 1 = 1.

How do you find sin⁡(75)\sin(75) using sum identities?

Express 75 as 45 + 30. Use the sine sum identity: sin⁡(45+30)=sin⁡(45)cos⁡(30)+cos⁡(45)sin⁡(30)\sin(45 + 30) = \sin(45)\cos(30) + \cos(45)\sin(30). Evaluate.

How to solve for cos⁡(x)\cos(x) given sin⁡(x)=35\sin(x) = \frac{3}{5} and xx is in the first quadrant?

  1. Use the Pythagorean identity: cos⁡2(x)=1−sin⁡2(x)\cos^2(x) = 1 - \sin^2(x). 2. Substitute: cos⁡2(x)=1−(35)2=1625\cos^2(x) = 1 - (\frac{3}{5})^2 = \frac{16}{25}. 3. Solve for cos⁡(x)\cos(x): cos⁡(x)=45\cos(x) = \frac{4}{5} (positive since x is in the first quadrant).

How to find sin⁡(2x)\sin(2x) if sin⁡(x)=0.6\sin(x) = 0.6 and cos⁡(x)=0.8\cos(x) = 0.8?

  1. Use the double-angle identity: sin⁡(2x)=2sin⁡(x)cos⁡(x)\sin(2x) = 2\sin(x)\cos(x). 2. Substitute: sin⁡(2x)=2(0.6)(0.8)\sin(2x) = 2(0.6)(0.8). 3. Calculate: sin⁡(2x)=0.96\sin(2x) = 0.96.

How to simplify cos⁡(a+b)−cos⁡(a−b)\cos(a + b) - \cos(a - b)?

  1. Expand using sum and difference identities: [cos(a)cos⁡(b)−sin⁡(a)sin⁡(b)]−[cos(a)cos⁡(b)+sin⁡(a)sin⁡(b)][cos(a)\cos(b) - \sin(a)\sin(b)] - [cos(a)\cos(b) + \sin(a)\sin(b)]. 2. Simplify: −2sin⁡(a)sin⁡(b)-2\sin(a)\sin(b).