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### Solve for x

```
Date: 10/26/2001 at 00:02:26
From: veronica
Subject: Trig equations

Solve over (0,2pi)

1. tan^2 x = 2sec x - 1    2. sin x - sqrt3 cos x = 1

Thank you.
```

```
Date: 10/26/2001 at 11:40:13
From: Doctor Rob
Subject: Re: Trig equations

Thanks for writing to Ask Dr. Math, Veronica.

1. Use the identity tan^2(x) + 1 = sec^2(x), and convert this into
a quadratic equation in sec(x). Solve it for sec(x). Then finish.

2. There is a general technique for equations of the form

a*sin(x) + b*cos(x) = c.

Divide by sqrt(a^2+b^2), and figure out the angle A such that

cos(A) = a/sqrt(a^2+b^2),
sin(A) = b/sqrt(a^2+b^2).

Then the equation will take the form

cos(A)*sin(x) + sin(A)*cos(x) = c/sqrt(a^2+b^2),
sin(x+A) = c/sqrt(a^2+b^2).

(Why?) That will allow you to figure out x + A, and thus x.

In your case, a = 1 and b = -sqrt(3), so sqrt(a^2+b^2) = 2.
Then cos(A) = 1/2, sin(A) = -sqrt(3)/2, so A = -60 degrees, or

- Doctor Rob, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Trigonometry

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