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Express Sine in Terms of Tangent

Even though each trigonometry function is perfectly wonderful, being able to express each trig function in terms of one of the other five trig functions is frequently to your advantage. For example, you may have some sine terms in an expression that you want to express in terms of tangent, so that all the functions match, making it easier to solve the equation.

To rewrite the sine function in terms of tangent, follow these steps:

  1. Start with the ratio identity involving sine, cosine, and tangent, and multiply each side by cosine to get the sine alone on the left.

    image0.png
  2. Replace cosine with its reciprocal function.

    image1.png
  3. Solve the Pythagorean identity tan2θ + 1 = sec2θ for secant.

    image2.png
  4. Replace the secant in the sine equation.

    image3.png
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