Solving Trigonometric Equations With Examples Example 3.3.3c: solving an equation involving tangent. solve the equation exactly: tan(θ − π 2) = 1, 0 ≤ θ <2π. solution. recall that the tangent function has a period of π. on the interval [0, π),and at the angle of π 4,the tangent has a value of 1. however, the angle we want is (θ − π 2). thus, if tan(π 4) = 1,then. We use some results and general solutions of the basic trigonometric equations to solve other trigonometric equations. these results are as follows: for any real numbers x and y, sin x = sin y implies x = nπ ( 1) n y, where n ∈ z. for any real numbers x and y, cos x = cos y implies x = 2nπ ± y, where n ∈ z.
Solving Trigonometric Equations Degrees Intomath We can set each factor equal to zero and solve. this is one example of recognizing algebraic patterns in trigonometric expressions or equations. another example is the difference of squares formula, \(a^2−b^2=(a−b)(a b)\), which is widely used in many areas other than mathematics, such as engineering, architecture, and physics. Example 6.6. solve the equation cos 3θ = 1 2. solution: the idea here is to solve for 3θ first, using the most general solution, and then divide that solution by 3. so since cos − 11 2 = π 3, there are two possible solutions for 3θ: 3θ = π 3 in qi and its reflection − 3θ = − π 3 around the x axis in qiv. Recall the rule that gives the format for stating all possible solutions for a function where the period is 2π: sinθ = sin(θ ± 2kπ) there are similar rules for indicating all possible solutions for the other trigonometric functions. solving trigonometric equations requires the same techniques as solving algebraic equations. Let’s just jump into the examples and see how to solve trig equations. example 1 solve 2cos(t) = √3. show solution. there’s really not a whole lot to do in solving this kind of trig equation. we first need to get the trig function on one side by itself. to do this all we need to do is divide both sides by 2.
How To Solve Trigonometric Equations Recall the rule that gives the format for stating all possible solutions for a function where the period is 2π: sinθ = sin(θ ± 2kπ) there are similar rules for indicating all possible solutions for the other trigonometric functions. solving trigonometric equations requires the same techniques as solving algebraic equations. Let’s just jump into the examples and see how to solve trig equations. example 1 solve 2cos(t) = √3. show solution. there’s really not a whole lot to do in solving this kind of trig equation. we first need to get the trig function on one side by itself. to do this all we need to do is divide both sides by 2. Substitute the trigonometric expression with a single variable, such as [latex]x [ latex] or [latex]u [ latex]. solve the equation the same way an algebraic equation would be solved. substitute the trigonometric expression back in for the variable in the resulting expressions. solve for the angle. Here are some examples of equations. 5(2 6) = 5(2) 5(6) √32 √42 = 3 4. x2 3x = 10. the first equation is true, the second is false, and the third equation is true only if x = 2 or x = − 5. when you solve an equation, you are finding the values of the variable that make the equation true. example 5.19.
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