Step 1 :Given the point (40,9) on the terminal side of an angle in standard position, we can find the values of the trigonometric functions of the angle.
Step 2 :The x-coordinate corresponds to the cosine of the angle, the y-coordinate corresponds to the sine of the angle, and the tangent of the angle is the y-coordinate divided by the x-coordinate.
Step 3 :The cotangent, secant, and cosecant are the reciprocals of the tangent, cosine, and sine respectively.
Step 4 :First, we calculate the distance from the origin to the point, which is \( r = \sqrt{x^2 + y^2} = \sqrt{40^2 + 9^2} = 41.0 \)
Step 5 :Then, we can calculate the trigonometric functions as follows:
Step 6 :\( \sin \theta = \frac{y}{r} = \frac{9}{41.0} = 0.22 \) (rounded to two decimal places)
Step 7 :\( \cos \theta = \frac{x}{r} = \frac{40}{41.0} = 0.98 \) (rounded to two decimal places)
Step 8 :\( \tan \theta = \frac{y}{x} = \frac{9}{40} = 0.23 \) (rounded to two decimal places)
Step 9 :\( \cot \theta = \frac{1}{\tan \theta} = \frac{1}{0.23} = 4.44 \) (rounded to two decimal places)
Step 10 :\( \sec \theta = \frac{1}{\cos \theta} = \frac{1}{0.98} = 1.03 \) (rounded to two decimal places)
Step 11 :\( \csc \theta = \frac{1}{\sin \theta} = \frac{1}{0.22} = 4.56 \) (rounded to two decimal places)
Step 12 :Final Answer:
Step 13 :\( \sin \theta = \boxed{0.22} \)
Step 14 :\( \cos \theta = \boxed{0.98} \)
Step 15 :\( \tan \theta = \boxed{0.23} \)
Step 16 :\( \cot \theta = \boxed{4.44} \)
Step 17 :\( \sec \theta = \boxed{1.03} \)
Step 18 :\( \csc \theta = \boxed{4.56} \)