Using the primary trigonometric formula explain why cos fada equals 1.4 has no solutions.
The primary trigonometric formula for cosine (cos) is:
cos θ = adjacent / hypotenuse
In this case, we are given that cos θ = 1.4. However, the range of values for cosine is -1 to 1. Since 1.4 is greater than 1, it is outside the valid range of the cosine function.
Therefore, there are no solutions for cos θ = 1.4.
The primary trigonometric formula involving the cosine function is:
cos(θ) = adjacent side / hypotenuse
In the given equation, cos(θ) = 1.4, we're looking for an angle (θ) whose cosine is equal to 1.4. In a right triangle, the value of the cosine function can only range from -1 to 1. So, any value greater than 1 or less than -1 is not possible.
Since the cosine function ranges from -1 to 1, and 1.4 exceeds this range, there are no solutions for the equation cos(θ) = 1.4.
To determine whether the equation cos θ = 1.4 has any solutions, we need to consider the range of values that cosine function can take.
The cosine function, cos θ, is defined as the ratio of the adjacent side to the hypotenuse in a right angle triangle, where θ is one of the acute angles. The range of values for cosine lies between -1 and 1 inclusive.
In this case, we have cos θ = 1.4. Since cosine can only take values between -1 and 1, the equation cos θ = 1.4 has no solutions.
To understand why this is the case using the primary trigonometric formula, we can express cosine in terms of the other trigonometric functions:
cos θ = 1/sin θ
Now, we can substitute this expression into the original equation:
1/sin θ = 1.4
To solve this equation for sin θ, we can cross-multiply:
1 = 1.4sin θ
sin θ = 1/1.4
sin θ ≈ 0.7143
However, the range of values for sine lies between -1 and 1. Since sin θ = 0.7143 is outside of this range, the equation sin θ = 0.7143 also has no solutions.
Therefore, using the primary trigonometric formula, we can conclude that the equation cos θ = 1.4 has no solutions.